#27107035

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

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #61867

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₄₇×#3119	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3119	D₄₇
hypocenter	C₄₇	#61867
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3119

1628 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)(64,68,70,69)(65,73,74,82)(66,77,76,81)(67,79,80,83)(71,86,85,87)(72,88,89,78)(75,92,91,93)(84,90,94,95);(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)(48,51,52,60)(49,54,55,63)(50,57,58,59)(53,61,56,62)(66,76)(71,85)(75,91)(77,81)(84,94)(86,87)(90,95)(92,93);(64,67)(65,72)(66,75)(68,79)(69,83)(70,80)(71,84)(73,88)(74,89)(76,91)(77,92)(78,82)(81,93)(85,94)(86,90)(87,95);(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61)(64,66)(65,71)(67,75)(68,81)(69,77)(70,76)(72,84)(73,87)(74,85)(78,90)(79,93)(80,91)(82,86)(83,92)(88,95)(89,94);(48,49)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65)(66,71)(67,78)(68,73)(69,82)(70,74)(72,79)(75,90)(76,85)(77,86)(80,88)(81,87)(83,89)(84,93)(91,95)(92,94)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1060325);