#27158513

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#135×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ #2957

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×#87604
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₄	C₂³×#87604
Fitting	#135×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₄×D₄
prime radical	#135×D₉₇	C₂
cosocle	C₄×D₉₇	C₂⁴
hypercenter	#135	#87604
hypocenter	C₉₇	#2957
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 129 points:

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

Polycyclic presentation:

2^7,97|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^22,e^b=eg,d^c=dg,h^c=h^22,h^d=h^96,h^f=h^96

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21287);