C₂×#20257901 #27123603

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#20257901

Splittings

Fitting	C₂×C₄₇×H(3,𝔽₄) ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3558

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×H(3,𝔽₄)	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₄₇
hypocenter	C₄₇	C₂×#3558
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

8122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 generators on 65 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)(51,55)(56,57);(50,53,54,52)(51,57,55,56)(58,60,61,64)(59,63,62,65);(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)(50,52,54,53)(51,56,55,57)(58,59)(60,63)(61,62)(64,65);(51,55)(52,53)(59,62)(63,65);(50,51)(52,57)(53,56)(54,55)(59,62)(60,64);(48,49)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076893);

C2×#20257901 #27123603