#27098752

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3342 ⋊ C₂
Sylow	C₄₇ ⋊ #61475

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3342	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3342	D₄₇
hypocenter	C₄₇	#61475
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3342

1900 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)(48,50,51,54)(49,53,52,55)(56,59,60,58)(57,65,62,64)(61,68,69,67)(63,71,66,70);(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)(49,52)(53,55)(56,58,60,59)(57,64,62,65)(61,67,69,68)(63,70,66,71);(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)(57,63)(61,69)(62,66)(64,70)(65,71)(67,68);(48,49)(50,53)(51,52)(54,55)(57,62)(58,59)(63,66)(67,68);(49,52)(53,55)(56,57,60,62)(58,64,59,65)(61,63,69,66)(67,70,68,71)

Polycyclic presentation:

2^8,47|a^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh,i^c=i^46,i^d=i^46,i^e=i^46

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1052042);