#27083026

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×B(4,𝔽₂) ⋊ C₂
Sylow	C₄₇ ⋊ #60399

Subgroups

	subgroup	G/H
center	C₂	C₂×#120×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×B(4,𝔽₂)	C₂
socle	C₂×C₄₇	C₂²×#120
hypercenter	C₂×B(4,𝔽₂)	D₄₇
hypocenter	C₄₇	#60399
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×B(4,𝔽₂)

5128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 752

A minimal faithful permutation representation — 5 generators on 63 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)(51,58)(54,61)(56,62)(60,63);(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)(49,54)(50,56)(52,58)(53,60)(55,61)(57,62)(59,63);(48,50)(49,53)(51,56)(52,57)(54,60)(55,59)(58,62)(61,63);(48,49)(50,53)(51,54)(52,55)(56,60)(57,59)(58,61)(62,63);(53,59)(60,63)

Polycyclic presentation:

2^8,47|b^a=bf,c^a=cg,g^b=gh,f^c=fh,e^d=eh,i^e=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, 1036316);