#27154936

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	#520×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3428

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator	C₂×C₉₇	C₂⁴×C₄
Fitting	#520×C₉₇	C₂
socle, cosocle	C₂²×C₉₇	C₂⁵
prime radical	C₂×#41×D₉₇	C₂
hypercenter	#520	D₉₇
hypocenter	C₉₇	#3428
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #520

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 117 points:

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

Polycyclic presentation:

2^7,97|a^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf,h^d=h^96,h^e=h^96

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 17710);