#72210652

order	6000
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₅² ⋊ #6732
derived	#72209598 ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	#72209598	C₂
Fitting, socle	C₅²	#6732
cosocle	C₅⋊D₅	C₂×A₅
LCS	G ▷ #72209598
UCS	C₁

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(2,𝔽₅), C₂×#72210652
maximal normal subgroup of	AGL(2,𝔽₅), C₂×#72210652
maximal quotient of	C₂×#72210652, #72277865

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

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