#27139950

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	Q₆₄×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ SD₁₂₈

Subgroups

	subgroup	G/H
center	C₂	#20260570
commutator, cosocle	C₃₂×C₉₇	C₂²
Frattini	C₃₂	C₂×D₉₇
Fitting	Q₆₄×C₉₇	C₂
socle	C₂×C₉₇	D₃₂
prime radical	D₃₁₀₄	C₂
hypercenter	Q₆₄	D₉₇
hypocenter	C₉₇	SD₁₂₈
LCS	G ▷ C₃₂×C₉₇ ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 161 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)(99,105)(100,111)(101,112)(102,117)(103,121)(106,107)(108,136)(109,138)(110,127)(113,144)(114,147)(115,128)(116,124)(118,152)(119,132)(120,125)(122,135)(123,126)(129,149)(130,148)(131,134)(133,154)(137,161)(139,140)(141,145)(142,143)(146,160)(150,157)(151,156)(153,159)(155,158);(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)(98,99,104,110)(100,105,115,127)(101,116,119,150)(102,120,122,153)(103,123,121,109)(106,128,131,111)(107,132,134,112)(108,135,137,117)(113,136,146,161)(114,138,145,126)(118,151,142,149)(124,155,157,140)(125,158,159,139)(129,160,156,144)(130,141,154,147)(133,143,148,152)

Polycyclic presentation:

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

Conjugacy classes & character table

947 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2724);