#27098700

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

Aliases

No aliases for this group.

Splittings

Fitting	D₄×Q₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #61473

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	D₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	D₄×Q₁₆	D₄₇
hypocenter	C₄₇	#61473
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₁₆

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:

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

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,d^b=df,e^b=eh,g^c=gh,i^e=i^46

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1051990);