#27106684

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

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #61852

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3556	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#3556	D₄₇
hypocenter	C₄₇	#61852
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #533 ◁ #3556

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 95 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(49,55)(53,56)(54,63)(61,62)(66,75)(70,84)(74,89)(76,91)(83,94)(85,92)(88,95)(90,93);(48,51,52,60)(49,54,55,63)(50,57,58,59)(53,61,56,62);(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)(64,67,69,79)(65,71,73,86)(66,74,75,89)(68,81,82,78)(70,83,84,94)(72,87,80,77)(76,93,91,90)(85,95,92,88);(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61)(64,66)(65,70)(67,74)(68,76)(69,75)(71,83)(72,85)(73,84)(77,88)(78,90)(79,89)(80,92)(81,93)(82,91)(86,94)(87,95);(48,49)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65,69,73)(66,70,75,84)(67,77,79,87)(68,80,82,72)(71,81,86,78)(74,88,89,95)(76,92,91,85)(83,93,94,90)

Polycyclic presentation:

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

Conjugacy classes & character table

923 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1059974);