#27140032

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	#322×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #1453

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	#322×C₉₇	C₂
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	#322	D₉₇
hypocenter	C₉₇	#1453
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ #322

190 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 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)(99,104)(100,105)(108,114)(111,117)(115,120)(118,121);(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)(98,100,102,104,103,105,101,99)(106,108)(107,111)(109,115)(110,114)(112,118)(113,117)(116,120)(119,121);(98,99,101,105,103,104,102,100)(106,107,109,112,110,113,116,119)(108,111,115,118,114,117,120,121)

Polycyclic presentation:

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

Conjugacy classes & character table

1976 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2806);