#27080674

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	Q₈×M₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #60192

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	Q₈×M₁₆×C₄₇	C₂
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	Q₈×M₁₆	D₄₇
hypocenter	C₄₇	#60192
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ Q₈×M₁₆

1882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 71 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)(57,62)(61,67)(63,69)(68,71);(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)(58,64)(61,67)(65,70)(68,71);(56,58,59,65,60,64,66,70)(57,61,63,68,62,67,69,71);(48,50,51,55)(49,52,53,54);(48,49,51,53)(50,54,55,52)(56,57,59,63,60,62,66,69)(58,61,65,68,64,67,70,71)

Polycyclic presentation:

2^8,47|a^2=g,b^2=f,c^2=fg,g^2=h,b^a=bf,e^a=eh,d^c=dh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

1238 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033964);