#27106758

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

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #61856

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3126	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₄₇×#540	C₂²
hypercenter	#3126	D₄₇
hypocenter	C₄₇	#61856
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3126

1612 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)(65,72)(66,76)(71,86)(73,82)(75,91)(77,93)(78,87)(92,94);(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)(70,74)(75,91)(77,93)(84,88)(85,89)(90,95)(92,94);(64,67,68,79)(65,71,72,86)(66,75,76,91)(69,83,81,80)(70,84,74,88)(73,87,82,78)(77,94,93,92)(85,90,89,95);(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61)(64,66,68,76)(65,70,72,74)(67,75,79,91)(69,77,81,93)(71,84,86,88)(73,85,82,89)(78,95,87,90)(80,92,83,94);(48,49)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65)(66,74)(67,78)(68,72)(69,82)(70,76)(71,80)(73,81)(75,90)(77,85)(79,87)(83,86)(84,94)(88,92)(89,93)(91,95)

Polycyclic presentation:

2^8,47|b^2=f,c^2=h,d^2=fh,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, 1060048);