#27080202

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	D₄×M₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #60141

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	D₄×M₁₆×C₄₇	C₂
prime radical	C₂×#15689632	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	D₄×M₁₆	D₄₇
hypocenter	C₄₇	#60141
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ D₄×M₁₆

2862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 1504

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

Polycyclic presentation:

2^8,47|a^2=g,c^2=g,g^2=h,b^a=bf,e^a=eh,d^c=dh,i^d=i^46,i^e=i^46

Conjugacy classes & character table

1238 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033492);