#27091452

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3275 ⋊ C₂
Sylow	C₄₇ ⋊ #61038

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3275	C₂
socle	C₂²×C₄₇	C₂³×D₄
hypercenter	#3275	D₄₇
hypocenter	C₄₇	#61038
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #3275

2790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 67 points:

(1,2)(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)(52,54,56,62)(53,60,58,65)(55,61,63,67)(57,64,59,66);(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)(53,59)(55,63)(57,58)(60,66)(61,67)(64,65);(49,51)(53,58)(57,59)(60,65)(64,66);(52,53)(54,60)(55,59)(56,58)(57,63)(61,66)(62,65)(64,67);(48,49)(50,51)(53,57)(55,63)(58,59)(60,64)(61,67)(65,66)

Polycyclic presentation:

2^8,47|e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh,i^d=i^46,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, 1044742);