#27151564

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

Aliases

No aliases for this group.

Splittings

Fitting	#478×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3081

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#478×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×D₄×D₉₇	C₂²
hypercenter	#478	D₉₇
hypocenter	C₉₇	#3081
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 121 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)(108,115)(111,119)(116,121)(118,120);(107,112)(109,117)(111,118)(113,114)(116,121)(119,120);(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)(98,100,101,105)(99,102,103,104)(106,108)(107,111)(109,116)(110,115)(112,118)(113,120)(114,119)(117,121);(98,99,101,103)(100,104,105,102)(106,107)(108,111)(109,112)(110,114)(113,117)(115,119)(116,118)(120,121)

Polycyclic presentation:

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

Conjugacy classes & character table

944 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 14338);