#27158301

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #2873

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87604
commutator	C₂×C₉₇	C₂²×C₄²
Fitting	#117×C₉₇	C₄
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	#117	#87604
hypocenter	C₉₇	#2873
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #117

564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 117 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)(103,108)(107,113)(109,115)(114,117);(104,110)(107,113)(111,116)(114,117);(98,99,100,101)(102,104)(103,107)(105,111)(106,110)(108,113)(109,114)(112,116)(115,117);(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)(102,103,105,109,106,108,112,115)(104,107,111,114,110,113,116,117)

Polycyclic presentation:

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

Conjugacy classes & character table

548 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21075);