C₂×#20251847 #27079746

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#20251847

Splittings

Fitting	C₂×C₄₇×#410 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#2883

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#410	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#13094070	C₂
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#410	D₄₇
hypocenter	C₄₇	C₂×#2883
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#410

5338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 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)(53,61)(56,63)(59,65)(62,64);(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)(50,53)(51,56)(52,59)(54,61)(55,62)(57,63)(58,64)(60,65);(50,52)(51,55)(53,59)(54,60)(56,62)(57,58)(61,65)(63,64);(50,51)(52,58,60,55)(53,56)(54,57)(59,64,65,62)(61,63);(48,49)

Conjugacy classes & character table

1094 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033036);

C2×#20251847 #27079746