#27098920

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

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #61484

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3523	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	#121×C₄₇	D₄
hypercenter	#3523	D₄₇
hypocenter	C₄₇	#61484
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #529 ◁ #3523

1750 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)(58,68)(62,66)(67,83)(69,85)(76,80)(77,81)(82,87)(84,86);(49,52)(53,55)(58,68)(62,66)(67,83)(69,85)(76,80)(77,81)(82,87)(84,86);(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)(56,59,60,71)(57,63,64,78)(58,67,68,83)(61,75,73,72)(62,76,66,80)(65,79,74,70)(69,86,85,84)(77,82,81,87);(48,49)(50,53)(51,52)(54,55)(56,58)(57,62)(59,67)(60,68)(61,69)(63,76)(64,66)(65,77)(70,87)(71,83)(72,84)(73,85)(74,81)(75,86)(78,80)(79,82);(49,52)(53,55)(56,57,60,64)(58,66,68,62)(59,70,71,79)(61,74,73,65)(63,75,78,72)(67,82,83,87)(69,77,85,81)(76,84,80,86)

Polycyclic presentation:

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

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1052210);