#27155880

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	#562×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3531

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂⁵
Fitting	#562×C₉₇	C₂
hypercenter	#562	D₉₇
hypocenter	C₉₇	#3531
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #562

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 5 generators on 129 points:

(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)(114,117,118,125)(115,121,120,127)(116,124,123,129)(119,126,122,128);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(99,105)(103,112)(104,108)(111,113)(115,120)(116,123)(121,127)(124,129);(98,101,102,110)(99,104,105,108)(100,106,107,109)(103,111,112,113)(116,123)(119,122)(124,129)(126,128);(98,100,102,107)(99,103,105,112)(101,109,110,106)(104,113,108,111)(114,116)(115,119)(117,124)(118,123)(120,122)(121,126)(125,129)(127,128);(98,99)(100,103)(101,108)(102,105)(104,110)(106,113)(107,112)(109,111)(114,115)(116,122)(117,121)(118,120)(119,123)(124,128)(125,127)(126,129)

Polycyclic presentation:

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

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 18654);