C₂×#20252788 #27083010

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#20252788

Splittings

Fitting	C₂×C₄₇×B(4,𝔽₂) ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3027

Subgroups

	subgroup	G/H
center	C₂²	#120×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×B(4,𝔽₂)	C₂
socle	C₂²×C₄₇	C₂×#120
hypercenter	C₂×B(4,𝔽₂)	D₄₇
hypocenter	C₄₇	C₂×#3027
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×B(4,𝔽₂)

5110 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036300);

C2×#20252788 #27083010