#27093227

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3322 ⋊ C₂
Sylow	C₄₇ ⋊ #61154

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3322	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3322	D₄₇
hypocenter	C₄₇	#61154
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3322

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 71 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)(56,59,61,60)(57,64,63,68)(58,67,66,69)(62,71,70,65);(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)(50,55)(52,54)(57,63)(62,70)(64,68)(65,71);(56,58)(57,62)(59,67)(60,69)(61,66)(63,70)(64,71)(65,68);(48,50)(49,52)(51,55)(53,54)(58,66)(62,70)(65,71)(67,69);(48,49,51,53)(50,54,55,52)(56,57)(58,65)(59,64)(60,68)(61,63)(62,67)(66,71)(69,70)

Polycyclic presentation:

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

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1046517);