C₂²×#15690885 #27158699

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#15690885

Splittings

Sylow

C₉₇ ⋊ C₂²×M₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#87604
commutator	C₂×C₉₇	C₂³×C₈
Frattini	C₄	C₂³×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle	C₂³×C₉₇	C₂×C₈
prime radical	C₂³×D₉₇	C₈
cosocle	#11715895	C₂⁴
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	C₂²×M₃₂
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 6208

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)(103,107)(108,113)(109,114)(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)(102,103,104,108,105,109,110,115,106,107,111,113,112,114,116,117);(100,101);(98,99)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21473);

C22×#15690885 #27158699