#27089011

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3080 ⋊ C₂
Sylow	C₄₇ ⋊ #60891

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3080	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#15689701	C₂²
hypercenter	#3080	D₄₇
hypocenter	C₄₇	#60891
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3080

1636 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)(66,76)(67,78)(70,83)(72,86)(77,90)(79,91)(81,84)(85,87);(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)(64,67)(65,72)(66,75)(68,79)(69,78)(70,82)(71,85)(73,87)(74,86)(76,89)(77,88)(80,91)(81,92)(83,94)(84,93)(90,95);(48,51,52,60)(49,54,55,63)(50,57,58,59)(53,61,56,62)(65,71)(68,80)(70,81)(72,85)(73,74)(77,90)(79,91)(82,92)(83,84)(86,87)(88,95)(93,94);(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)(48,50,52,58)(49,53,55,56)(51,59,60,57)(54,62,63,61)(64,66,69,76)(65,70,74,83)(67,75,78,89)(68,77,80,90)(71,81,73,84)(72,82,86,94)(79,88,91,95)(85,92,87,93);(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)(48,49)(50,56)(51,54)(52,55)(53,58)(57,62)(59,61)(60,63)(64,65,69,74)(66,70,76,83)(67,72,78,86)(68,71,80,73)(75,82,89,94)(77,81,90,84)(79,85,91,87)(88,92,95,93)

Polycyclic presentation:

2^8,47|a^2=h,b^2=fh,c^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,i^a=i^46,c^b=cf,e^b=eh,i^b=i^46,g^c=gh,e^d=eh,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, 1042301);