Finite Group Databasebeta

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C₂×H(13,𝔽₂) #72290091

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Subgroups

37559530483961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 13 generators on 130 points:

(4,11)(12,32)(13,33)(14,34)(15,35)(16,36)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(112,123)(113,124)(114,125)(115,126)(116,127)(128,130);(3,4)(5,12)(6,13)(7,14)(8,15)(9,16)(10,11)(17,32)(18,37)(19,38)(20,39)(21,40)(22,33)(23,41)(24,42)(25,43)(26,34)(27,44)(28,45)(29,35)(30,46)(31,36)(47,67)(48,68)(49,69)(50,70)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,71)(58,72)(59,73)(60,83)(61,84)(62,85)(63,74)(64,75)(65,86)(66,76)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,112)(94,113)(95,114)(96,115)(97,108)(98,109)(99,110)(100,116)(101,111)(117,123)(118,124)(119,125)(120,126)(121,128)(122,127)(129,130);(5,17)(12,32)(18,47)(19,48)(20,49)(21,50)(37,67)(38,68)(39,69)(40,70)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(93,117)(94,118)(95,119)(96,120)(112,123)(113,124)(114,125)(115,126)(121,129)(128,130);(3,5)(4,12)(6,18)(7,19)(8,20)(9,21)(10,17)(11,32)(13,37)(14,38)(15,39)(16,40)(22,47)(23,51)(24,52)(25,53)(26,48)(27,54)(28,55)(29,49)(30,56)(31,50)(33,67)(34,68)(35,69)(36,70)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(57,87)(58,88)(59,89)(60,93)(61,94)(62,95)(63,90)(64,91)(65,96)(66,92)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(83,112)(84,113)(85,114)(86,115)(97,117)(98,118)(99,119)(100,121)(101,120)(108,123)(109,124)(110,125)(111,126)(116,128)(122,129)(127,130);(6,22)(13,33)(18,47)(23,57)(24,58)(25,59)(37,67)(41,71)(42,72)(43,73)(51,87)(52,88)(53,89)(60,97)(61,98)(62,99)(77,102)(78,103)(79,104)(83,108)(84,109)(85,110)(93,117)(94,118)(95,119)(100,122)(112,123)(113,124)(114,125)(116,127)(121,129)(128,130);(3,6)(4,13)(5,18)(7,23)(8,24)(9,25)(10,22)(11,33)(12,37)(14,41)(15,42)(16,43)(17,47)(19,51)(20,52)(21,53)(26,57)(27,60)(28,61)(29,58)(30,62)(31,59)(32,67)(34,71)(35,72)(36,73)(38,77)(39,78)(40,79)(44,83)(45,84)(46,85)(48,87)(49,88)(50,89)(54,93)(55,94)(56,95)(63,97)(64,98)(65,100)(66,99)(68,102)(69,103)(70,104)(74,108)(75,109)(76,110)(80,112)(81,113)(82,114)(86,116)(90,117)(91,118)(92,119)(96,121)(101,122)(105,123)(106,124)(107,125)(111,127)(115,128)(120,129)(126,130);(7,26)(14,34)(19,48)(23,57)(27,63)(28,64)(38,68)(41,71)(44,74)(45,75)(51,87)(54,90)(55,91)(60,97)(61,98)(65,101)(77,102)(80,105)(81,106)(83,108)(84,109)(86,111)(93,117)(94,118)(96,120)(100,122)(112,123)(113,124)(115,126)(116,127)(121,129)(128,130);(3,7)(4,14)(5,19)(6,23)(8,27)(9,28)(10,26)(11,34)(12,38)(13,41)(15,44)(16,45)(17,48)(18,51)(20,54)(21,55)(22,57)(24,60)(25,61)(29,63)(30,65)(31,64)(32,68)(33,71)(35,74)(36,75)(37,77)(39,80)(40,81)(42,83)(43,84)(46,86)(47,87)(49,90)(50,91)(52,93)(53,94)(56,96)(58,97)(59,98)(62,100)(66,101)(67,102)(69,105)(70,106)(72,108)(73,109)(76,111)(78,112)(79,113)(82,115)(85,116)(88,117)(89,118)(92,120)(95,121)(99,122)(103,123)(104,124)(107,126)(110,127)(114,128)(119,129)(125,130);(8,29)(15,35)(20,49)(24,58)(27,63)(30,66)(39,69)(42,72)(44,74)(46,76)(52,88)(54,90)(56,92)(60,97)(62,99)(65,101)(78,103)(80,105)(82,107)(83,108)(85,110)(86,111)(93,117)(95,119)(96,120)(100,122)(112,123)(114,125)(115,126)(116,127)(121,129)(128,130);(3,8)(4,15)(5,20)(6,24)(7,27)(9,30)(10,29)(11,35)(12,39)(13,42)(14,44)(16,46)(17,49)(18,52)(19,54)(21,56)(22,58)(23,60)(25,62)(26,63)(28,65)(31,66)(32,69)(33,72)(34,74)(36,76)(37,78)(38,80)(40,82)(41,83)(43,85)(45,86)(47,88)(48,90)(50,92)(51,93)(53,95)(55,96)(57,97)(59,99)(61,100)(64,101)(67,103)(68,105)(70,107)(71,108)(73,110)(75,111)(77,112)(79,114)(81,115)(84,116)(87,117)(89,119)(91,120)(94,121)(98,122)(102,123)(104,125)(106,126)(109,127)(113,128)(118,129)(124,130);(9,31)(16,36)(21,50)(25,59)(28,64)(30,66)(40,70)(43,73)(45,75)(46,76)(53,89)(55,91)(56,92)(61,98)(62,99)(65,101)(79,104)(81,106)(82,107)(84,109)(85,110)(86,111)(94,118)(95,119)(96,120)(100,122)(113,124)(114,125)(115,126)(116,127)(121,129)(128,130);(3,9)(4,16)(5,21)(6,25)(7,28)(8,30)(10,31)(11,36)(12,40)(13,43)(14,45)(15,46)(17,50)(18,53)(19,55)(20,56)(22,59)(23,61)(24,62)(26,64)(27,65)(29,66)(32,70)(33,73)(34,75)(35,76)(37,79)(38,81)(39,82)(41,84)(42,85)(44,86)(47,89)(48,91)(49,92)(51,94)(52,95)(54,96)(57,98)(58,99)(60,100)(63,101)(67,104)(68,106)(69,107)(71,109)(72,110)(74,111)(77,113)(78,114)(80,115)(83,116)(87,118)(88,119)(90,120)(93,121)(97,122)(102,124)(103,125)(105,126)(108,127)(112,128)(117,129)(123,130);(1,2)

Conjugacy classes & character table

8194 conjugacy classes. The character table is too large to display.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,11)(12,32)(13,33)(14,34)(15,35)(16,36)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(112,123)(113,124)(114,125)(115,126)(116,127)(128,130), (3,4)(5,12)(6,13)(7,14)(8,15)(9,16)(10,11)(17,32)(18,37)(19,38)(20,39)(21,40)(22,33)(23,41)(24,42)(25,43)(26,34)(27,44)(28,45)(29,35)(30,46)(31,36)(47,67)(48,68)(49,69)(50,70)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,71)(58,72)(59,73)(60,83)(61,84)(62,85)(63,74)(64,75)(65,86)(66,76)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,112)(94,113)(95,114)(96,115)(97,108)(98,109)(99,110)(100,116)(101,111)(117,123)(118,124)(119,125)(120,126)(121,128)(122,127)(129,130), (5,17)(12,32)(18,47)(19,48)(20,49)(21,50)(37,67)(38,68)(39,69)(40,70)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(93,117)(94,118)(95,119)(96,120)(112,123)(113,124)(114,125)(115,126)(121,129)(128,130), (3,5)(4,12)(6,18)(7,19)(8,20)(9,21)(10,17)(11,32)(13,37)(14,38)(15,39)(16,40)(22,47)(23,51)(24,52)(25,53)(26,48)(27,54)(28,55)(29,49)(30,56)(31,50)(33,67)(34,68)(35,69)(36,70)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(57,87)(58,88)(59,89)(60,93)(61,94)(62,95)(63,90)(64,91)(65,96)(66,92)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(83,112)(84,113)(85,114)(86,115)(97,117)(98,118)(99,119)(100,121)(101,120)(108,123)(109,124)(110,125)(111,126)(116,128)(122,129)(127,130), (6,22)(13,33)(18,47)(23,57)(24,58)(25,59)(37,67)(41,71)(42,72)(43,73)(51,87)(52,88)(53,89)(60,97)(61,98)(62,99)(77,102)(78,103)(79,104)(83,108)(84,109)(85,110)(93,117)(94,118)(95,119)(100,122)(112,123)(113,124)(114,125)(116,127)(121,129)(128,130), (3,6)(4,13)(5,18)(7,23)(8,24)(9,25)(10,22)(11,33)(12,37)(14,41)(15,42)(16,43)(17,47)(19,51)(20,52)(21,53)(26,57)(27,60)(28,61)(29,58)(30,62)(31,59)(32,67)(34,71)(35,72)(36,73)(38,77)(39,78)(40,79)(44,83)(45,84)(46,85)(48,87)(49,88)(50,89)(54,93)(55,94)(56,95)(63,97)(64,98)(65,100)(66,99)(68,102)(69,103)(70,104)(74,108)(75,109)(76,110)(80,112)(81,113)(82,114)(86,116)(90,117)(91,118)(92,119)(96,121)(101,122)(105,123)(106,124)(107,125)(111,127)(115,128)(120,129)(126,130), (7,26)(14,34)(19,48)(23,57)(27,63)(28,64)(38,68)(41,71)(44,74)(45,75)(51,87)(54,90)(55,91)(60,97)(61,98)(65,101)(77,102)(80,105)(81,106)(83,108)(84,109)(86,111)(93,117)(94,118)(96,120)(100,122)(112,123)(113,124)(115,126)(116,127)(121,129)(128,130), (3,7)(4,14)(5,19)(6,23)(8,27)(9,28)(10,26)(11,34)(12,38)(13,41)(15,44)(16,45)(17,48)(18,51)(20,54)(21,55)(22,57)(24,60)(25,61)(29,63)(30,65)(31,64)(32,68)(33,71)(35,74)(36,75)(37,77)(39,80)(40,81)(42,83)(43,84)(46,86)(47,87)(49,90)(50,91)(52,93)(53,94)(56,96)(58,97)(59,98)(62,100)(66,101)(67,102)(69,105)(70,106)(72,108)(73,109)(76,111)(78,112)(79,113)(82,115)(85,116)(88,117)(89,118)(92,120)(95,121)(99,122)(103,123)(104,124)(107,126)(110,127)(114,128)(119,129)(125,130), (8,29)(15,35)(20,49)(24,58)(27,63)(30,66)(39,69)(42,72)(44,74)(46,76)(52,88)(54,90)(56,92)(60,97)(62,99)(65,101)(78,103)(80,105)(82,107)(83,108)(85,110)(86,111)(93,117)(95,119)(96,120)(100,122)(112,123)(114,125)(115,126)(116,127)(121,129)(128,130), (3,8)(4,15)(5,20)(6,24)(7,27)(9,30)(10,29)(11,35)(12,39)(13,42)(14,44)(16,46)(17,49)(18,52)(19,54)(21,56)(22,58)(23,60)(25,62)(26,63)(28,65)(31,66)(32,69)(33,72)(34,74)(36,76)(37,78)(38,80)(40,82)(41,83)(43,85)(45,86)(47,88)(48,90)(50,92)(51,93)(53,95)(55,96)(57,97)(59,99)(61,100)(64,101)(67,103)(68,105)(70,107)(71,108)(73,110)(75,111)(77,112)(79,114)(81,115)(84,116)(87,117)(89,119)(91,120)(94,121)(98,122)(102,123)(104,125)(106,126)(109,127)(113,128)(118,129)(124,130), (9,31)(16,36)(21,50)(25,59)(28,64)(30,66)(40,70)(43,73)(45,75)(46,76)(53,89)(55,91)(56,92)(61,98)(62,99)(65,101)(79,104)(81,106)(82,107)(84,109)(85,110)(86,111)(94,118)(95,119)(96,120)(100,122)(113,124)(114,125)(115,126)(116,127)(121,129)(128,130), (3,9)(4,16)(5,21)(6,25)(7,28)(8,30)(10,31)(11,36)(12,40)(13,43)(14,45)(15,46)(17,50)(18,53)(19,55)(20,56)(22,59)(23,61)(24,62)(26,64)(27,65)(29,66)(32,70)(33,73)(34,75)(35,76)(37,79)(38,81)(39,82)(41,84)(42,85)(44,86)(47,89)(48,91)(49,92)(51,94)(52,95)(54,96)(57,98)(58,99)(60,100)(63,101)(67,104)(68,106)(69,107)(71,109)(72,110)(74,111)(77,113)(78,114)(80,115)(83,116)(87,118)(88,119)(90,120)(93,121)(97,122)(102,124)(103,125)(105,126)(108,127)(112,128)(117,129)(123,130), (1,2));