#27113141

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3121 ⋊ C₂
Sylow	C₄₇ ⋊ #62161

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3121	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3121	D₄₇
hypocenter	C₄₇	#62161
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3121

1646 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)(48,51,52,60)(49,54,55,59)(50,58,57,63)(53,61,56,62)(66,77)(71,86)(75,91)(76,78)(84,95)(85,92)(89,94)(90,93);(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)(50,57)(53,56)(58,63)(61,62)(64,68,70,69)(65,73,74,82)(66,76,77,78)(67,80,81,83)(71,85,86,92)(72,87,88,79)(75,90,91,93)(84,94,95,89);(64,67,70,81)(65,72,74,88)(66,75,77,91)(68,80,69,83)(71,84,86,95)(73,87,82,79)(76,90,78,93)(85,94,92,89);(48,50)(49,53)(51,58)(52,57)(54,61)(55,56)(59,62)(60,63)(64,66)(65,71)(67,75)(68,76)(69,78)(70,77)(72,84)(73,85)(74,86)(79,89)(80,90)(81,91)(82,92)(83,93)(87,94)(88,95);(48,49,52,55)(50,56,57,53)(51,59,60,54)(58,61,63,62)(64,65)(66,71)(67,79)(68,73)(69,82)(70,74)(72,80)(75,89)(76,85)(77,86)(78,92)(81,87)(83,88)(84,90)(91,94)(93,95)

Polycyclic presentation:

2^8,47|a^2=f,c^2=h,d^2=h,e^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^d=i^46

Conjugacy classes & character table

785 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1066431);