C₂×#20256586 #27122871

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#20256586

Splittings

Fitting	C₂×C₄₇×#518 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3428

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator	C₂×C₄₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#518	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂³×#10625352	C₂
cosocle	C₂²×C₄₇	C₂⁶
hypercenter	C₂×#518	D₄₇
hypocenter	C₄₇	C₂×#3428
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#518

11534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 generators on 69 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)(54,57,58,65)(55,61,60,67)(56,64,63,69)(59,66,62,68);(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)(55,60)(59,62)(61,67)(66,68);(56,63)(59,62)(64,69)(66,68);(54,56)(55,59)(57,64)(58,63)(60,62)(61,66)(65,69)(67,68);(50,51,52,53)(54,55)(56,62)(57,61)(58,60)(59,63)(64,68)(65,67)(66,69);(48,49)

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076161);

C2×#20256586 #27122871