Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| general orthogonal | GO+(4,𝔽4) |
| special orthogonal | SO+(4,𝔽4) |
| socle, derived | A52 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A52 | C2 |
| LCS | G ▷ A52 | |
| UCS | C1 | |
103 conjugacy classes of subgroups; only the 3 normal subgroups are shown.
This group has 103 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,34)(33,35)(36,38)(37,39)(40,42)(41,43)(44,46)(45,47)(48,51)(49,50)(52,55)(53,54)(56,59)(57,58)(60,63)(61,62)(64,68)(65,69)(66,70)(67,71)(72,76)(73,77)(74,78)(75,79)(80,85)(81,84)(82,87)(83,86)(88,93)(89,92)(90,95)(91,94)(96,102)(97,103)(98,100)(99,101)(104,110)(105,111)(106,108)(107,109)(112,119)(113,118)(114,117)(115,116)(120,127)(121,126)(122,125)(123,124)(128,136)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(144,153)(145,152)(146,155)(147,154)(148,157)(149,156)(150,159)(151,158)(160,170)(161,171)(162,168)(163,169)(164,174)(165,175)(166,172)(167,173)(176,187)(177,186)(178,185)(179,184)(180,191)(181,190)(182,189)(183,188)(192,204)(193,205)(194,206)(195,207)(196,200)(197,201)(198,202)(199,203)(208,221)(209,220)(210,223)(211,222)(212,217)(213,216)(214,219)(215,218)(224,238)(225,239)(226,236)(227,237)(228,234)(229,235)(230,232)(231,233)(240,255)(241,254)(242,253)(243,252)(244,251)(245,250)(246,249)(247,248);(16,18)(17,19)(20,22)(21,23)(24,26)(25,27)(28,30)(29,31)(32,35)(33,34)(36,39)(37,38)(40,43)(41,42)(44,47)(45,46)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(80,90)(81,91)(82,88)(83,89)(84,94)(85,95)(86,92)(87,93)(96,107)(97,106)(98,105)(99,104)(100,111)(101,110)(102,109)(103,108)(112,121)(113,120)(114,123)(115,122)(116,125)(117,124)(118,127)(119,126)(128,140)(129,141)(130,142)(131,143)(132,136)(133,137)(134,138)(135,139)(144,158)(145,159)(146,156)(147,157)(148,154)(149,155)(150,152)(151,153)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(176,189)(177,188)(178,191)(179,190)(180,185)(181,184)(182,187)(183,186)(192,196)(193,197)(194,198)(195,199)(200,204)(201,205)(202,206)(203,207)(208,214)(209,215)(210,212)(211,213)(216,222)(217,223)(218,220)(219,221)(224,231)(225,230)(226,229)(227,228)(232,239)(233,238)(234,237)(235,236)(240,245)(241,244)(242,247)(243,246)(248,253)(249,252)(250,255)(251,254);(4,5)(6,7)(8,10)(9,11)(12,15)(13,14)(20,21)(22,23)(24,26)(25,27)(28,31)(29,30)(36,37)(38,39)(40,42)(41,43)(44,47)(45,46)(52,53)(54,55)(56,58)(57,59)(60,63)(61,62)(64,80)(65,81)(66,82)(67,83)(68,85)(69,84)(70,87)(71,86)(72,90)(73,91)(74,88)(75,89)(76,95)(77,94)(78,93)(79,92)(96,112)(97,113)(98,114)(99,115)(100,117)(101,116)(102,119)(103,118)(104,122)(105,123)(106,120)(107,121)(108,127)(109,126)(110,125)(111,124)(128,160)(129,161)(130,162)(131,163)(132,165)(133,164)(134,167)(135,166)(136,170)(137,171)(138,168)(139,169)(140,175)(141,174)(142,173)(143,172)(144,176)(145,177)(146,178)(147,179)(148,181)(149,180)(150,183)(151,182)(152,186)(153,187)(154,184)(155,185)(156,191)(157,190)(158,189)(159,188)(192,240)(193,241)(194,242)(195,243)(196,245)(197,244)(198,247)(199,246)(200,250)(201,251)(202,248)(203,249)(204,255)(205,254)(206,253)(207,252)(208,224)(209,225)(210,226)(211,227)(212,229)(213,228)(214,231)(215,230)(216,234)(217,235)(218,232)(219,233)(220,239)(221,238)(222,237)(223,236);(4,6)(5,7)(8,11)(9,10)(12,13)(14,15)(20,22)(21,23)(24,27)(25,26)(28,29)(30,31)(36,38)(37,39)(40,43)(41,42)(44,45)(46,47)(52,54)(53,55)(56,59)(57,58)(60,61)(62,63)(64,96)(65,97)(66,98)(67,99)(68,102)(69,103)(70,100)(71,101)(72,107)(73,106)(74,105)(75,104)(76,109)(77,108)(78,111)(79,110)(80,112)(81,113)(82,114)(83,115)(84,118)(85,119)(86,116)(87,117)(88,123)(89,122)(90,121)(91,120)(92,125)(93,124)(94,127)(95,126)(128,176)(129,177)(130,178)(131,179)(132,182)(133,183)(134,180)(135,181)(136,187)(137,186)(138,185)(139,184)(140,189)(141,188)(142,191)(143,190)(144,160)(145,161)(146,162)(147,163)(148,166)(149,167)(150,164)(151,165)(152,171)(153,170)(154,169)(155,168)(156,173)(157,172)(158,175)(159,174)(192,208)(193,209)(194,210)(195,211)(196,214)(197,215)(198,212)(199,213)(200,219)(201,218)(202,217)(203,216)(204,221)(205,220)(206,223)(207,222)(224,240)(225,241)(226,242)(227,243)(228,246)(229,247)(230,244)(231,245)(232,251)(233,250)(234,249)(235,248)(236,253)(237,252)(238,255)(239,254);(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,34)(33,35)(36,38)(37,39)(40,42)(41,43)(44,46)(45,47)(48,51)(49,50)(52,55)(53,54)(56,59)(57,58)(60,63)(61,62)(64,68)(65,69)(66,70)(67,71)(72,76)(73,77)(74,78)(75,79)(80,85)(81,84)(82,87)(83,86)(88,93)(89,92)(90,95)(91,94)(96,102)(97,103)(98,100)(99,101)(104,110)(105,111)(106,108)(107,109)(112,119)(113,118)(114,117)(115,116)(120,127)(121,126)(122,125)(123,124)(128,136)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(144,153)(145,152)(146,155)(147,154)(148,157)(149,156)(150,159)(151,158)(160,170)(161,171)(162,168)(163,169)(164,174)(165,175)(166,172)(167,173)(176,187)(177,186)(178,185)(179,184)(180,191)(181,190)(182,189)(183,188)(192,204)(193,205)(194,206)(195,207)(196,200)(197,201)(198,202)(199,203)(208,221)(209,220)(210,223)(211,222)(212,217)(213,216)(214,219)(215,218)(224,238)(225,239)(226,236)(227,237)(228,234)(229,235)(230,232)(231,233)(240,255)(241,254)(242,253)(243,252)(244,251)(245,250)(246,249)(247,248);(16,18)(17,19)(20,22)(21,23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)(28,92)(29,125)(30,110)(31,79)(35,51)(36,164)(37,133)(38,150)(39,183)(40,232)(41,201)(42,218)(43,251)(44,108)(45,77)(46,94)(47,127)(52,180)(53,149)(54,134)(55,167)(56,248)(57,217)(58,202)(59,235)(60,124)(61,93)(62,78)(63,111)(65,97)(66,114)(67,83)(68,196)(69,229)(70,246)(71,215)(72,136)(73,169)(74,186)(75,155)(81,113)(82,98)(84,212)(85,245)(86,230)(87,199)(88,152)(89,185)(90,170)(91,139)(99,115)(100,228)(101,197)(102,214)(103,247)(104,168)(105,137)(106,154)(107,187)(116,244)(117,213)(118,198)(119,231)(120,184)(121,153)(122,138)(123,171)(129,161)(130,178)(131,147)(140,204)(141,237)(142,254)(143,223)(145,177)(146,162)(156,220)(157,253)(158,238)(159,207)(163,179)(172,236)(173,205)(174,222)(175,255)(188,252)(189,221)(190,206)(191,239)(193,225)(194,242)(195,211)(209,241)(210,226)(227,243);(4,16)(5,17)(6,18)(7,19)(8,32)(9,33)(10,34)(11,35)(12,48)(13,49)(14,50)(15,51)(24,36)(25,37)(26,38)(27,39)(28,52)(29,53)(30,54)(31,55)(44,56)(45,57)(46,58)(47,59)(68,80)(69,81)(70,82)(71,83)(72,96)(73,97)(74,98)(75,99)(76,112)(77,113)(78,114)(79,115)(88,100)(89,101)(90,102)(91,103)(92,116)(93,117)(94,118)(95,119)(108,120)(109,121)(110,122)(111,123)(132,144)(133,145)(134,146)(135,147)(136,160)(137,161)(138,162)(139,163)(140,176)(141,177)(142,178)(143,179)(152,164)(153,165)(154,166)(155,167)(156,180)(157,181)(158,182)(159,183)(172,184)(173,185)(174,186)(175,187)(196,208)(197,209)(198,210)(199,211)(200,224)(201,225)(202,226)(203,227)(204,240)(205,241)(206,242)(207,243)(216,228)(217,229)(218,230)(219,231)(220,244)(221,245)(222,246)(223,247)(236,248)(237,249)(238,250)(239,251)20 conjugacy classes — view the full character table.
| maximal subgroup of | #72219552 |
| maximal normal subgroup of | #72219552 |
| maximal quotient of | #72219550 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,34)(33,35)(36,38)(37,39)(40,42)(41,43)(44,46)(45,47)(48,51)(49,50)(52,55)(53,54)(56,59)(57,58)(60,63)(61,62)(64,68)(65,69)(66,70)(67,71)(72,76)(73,77)(74,78)(75,79)(80,85)(81,84)(82,87)(83,86)(88,93)(89,92)(90,95)(91,94)(96,102)(97,103)(98,100)(99,101)(104,110)(105,111)(106,108)(107,109)(112,119)(113,118)(114,117)(115,116)(120,127)(121,126)(122,125)(123,124)(128,136)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(144,153)(145,152)(146,155)(147,154)(148,157)(149,156)(150,159)(151,158)(160,170)(161,171)(162,168)(163,169)(164,174)(165,175)(166,172)(167,173)(176,187)(177,186)(178,185)(179,184)(180,191)(181,190)(182,189)(183,188)(192,204)(193,205)(194,206)(195,207)(196,200)(197,201)(198,202)(199,203)(208,221)(209,220)(210,223)(211,222)(212,217)(213,216)(214,219)(215,218)(224,238)(225,239)(226,236)(227,237)(228,234)(229,235)(230,232)(231,233)(240,255)(241,254)(242,253)(243,252)(244,251)(245,250)(246,249)(247,248), (16,18)(17,19)(20,22)(21,23)(24,26)(25,27)(28,30)(29,31)(32,35)(33,34)(36,39)(37,38)(40,43)(41,42)(44,47)(45,46)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(80,90)(81,91)(82,88)(83,89)(84,94)(85,95)(86,92)(87,93)(96,107)(97,106)(98,105)(99,104)(100,111)(101,110)(102,109)(103,108)(112,121)(113,120)(114,123)(115,122)(116,125)(117,124)(118,127)(119,126)(128,140)(129,141)(130,142)(131,143)(132,136)(133,137)(134,138)(135,139)(144,158)(145,159)(146,156)(147,157)(148,154)(149,155)(150,152)(151,153)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(176,189)(177,188)(178,191)(179,190)(180,185)(181,184)(182,187)(183,186)(192,196)(193,197)(194,198)(195,199)(200,204)(201,205)(202,206)(203,207)(208,214)(209,215)(210,212)(211,213)(216,222)(217,223)(218,220)(219,221)(224,231)(225,230)(226,229)(227,228)(232,239)(233,238)(234,237)(235,236)(240,245)(241,244)(242,247)(243,246)(248,253)(249,252)(250,255)(251,254), (4,5)(6,7)(8,10)(9,11)(12,15)(13,14)(20,21)(22,23)(24,26)(25,27)(28,31)(29,30)(36,37)(38,39)(40,42)(41,43)(44,47)(45,46)(52,53)(54,55)(56,58)(57,59)(60,63)(61,62)(64,80)(65,81)(66,82)(67,83)(68,85)(69,84)(70,87)(71,86)(72,90)(73,91)(74,88)(75,89)(76,95)(77,94)(78,93)(79,92)(96,112)(97,113)(98,114)(99,115)(100,117)(101,116)(102,119)(103,118)(104,122)(105,123)(106,120)(107,121)(108,127)(109,126)(110,125)(111,124)(128,160)(129,161)(130,162)(131,163)(132,165)(133,164)(134,167)(135,166)(136,170)(137,171)(138,168)(139,169)(140,175)(141,174)(142,173)(143,172)(144,176)(145,177)(146,178)(147,179)(148,181)(149,180)(150,183)(151,182)(152,186)(153,187)(154,184)(155,185)(156,191)(157,190)(158,189)(159,188)(192,240)(193,241)(194,242)(195,243)(196,245)(197,244)(198,247)(199,246)(200,250)(201,251)(202,248)(203,249)(204,255)(205,254)(206,253)(207,252)(208,224)(209,225)(210,226)(211,227)(212,229)(213,228)(214,231)(215,230)(216,234)(217,235)(218,232)(219,233)(220,239)(221,238)(222,237)(223,236), 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