#27144056

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #2264

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₅₅₂
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	M₆₄×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₁₆
prime radical	#20261167	C₂
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#2264
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 161 points:

(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)(99,106)(105,127)(107,130)(108,131)(109,125)(110,137)(115,136)(121,155)(128,157)(129,135)(132,158)(133,146)(134,159)(138,145)(142,161)(147,153);(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)(98,100,104,114)(99,105,106,127)(101,116,120,150)(102,122,124,156)(103,126,123,113)(107,132,130,158)(108,134,131,159)(109,135,125,129)(110,136,137,115)(111,139,141,118)(112,143,144,117)(119,154,152,149)(121,128,155,157)(133,161,146,142)(138,147,145,153)(140,151,148,160);(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)(98,99)(100,110)(101,115)(102,121)(103,125)(104,106)(105,111)(107,118)(108,117)(109,123)(112,142)(113,145)(114,137)(116,134)(119,153)(120,136)(122,132)(124,155)(126,138)(127,141)(128,149)(129,148)(130,139)(131,143)(133,160)(135,140)(144,161)(146,151)(147,152)(150,159)(154,157)(156,158)

Polycyclic presentation:

2^7,97|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^96,d^b=de,e^b=ef,f^b=fg,h^b=h^96

Conjugacy classes & character table

1946 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 6830);