#27099086

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3127 ⋊ C₂
Sylow	C₄₇ ⋊ #61491

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3127	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3127	D₄₇
hypocenter	C₄₇	#61491
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #3127

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 87 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)(48,50,51,54)(49,53,52,55)(58,67)(62,77)(66,70)(68,84)(76,79)(78,85)(82,87)(83,86);(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)(49,52)(53,55);(56,59,61,72)(57,63,65,80)(58,66,67,70)(60,74,75,71)(62,76,77,79)(64,81,73,69)(68,86,84,83)(78,87,85,82);(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)(50,53)(51,52)(54,55)(56,58)(57,62)(59,70)(60,68)(61,67)(63,79)(64,78)(65,77)(66,72)(69,87)(71,86)(73,85)(74,83)(75,84)(76,80)(81,82);(49,52)(53,55)(56,57,61,65)(58,62,67,77)(59,69,72,81)(60,73,75,64)(63,74,80,71)(66,82,70,87)(68,85,84,78)(76,86,79,83)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1052376);