#27106911

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3360 ⋊ C₂
Sylow	C₄₇ ⋊ #61862

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3360	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#15689265	C₂²
hypercenter	#3360	D₄₇
hypocenter	C₄₇	#61862
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #3360

1614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

854 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1060201);