#27158133

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	Q₃₂×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ #2176

Subgroups

	subgroup	G/H
center	C₂	D₈×#87604
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₈	C₂²×#87604
Fitting	Q₃₂×C₉₇	C₄
socle	C₂×C₉₇	C₄×D₈
prime radical	C₂×D₉₇	C₂×D₈
cosocle	C₈×D₉₇	C₂³
hypercenter	Q₃₂	#87604
hypocenter	C₉₇	#2176
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 129 points:

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

Polycyclic presentation:

2^7,97|a^2=e,b^2=g,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^75,d^b=df,f^b=fg,h^c=h^22,h^e=h^96

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20907);