C₂×#20253011 #27083429

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

Aliases

direct product

C₂×#20253011

Splittings

Fitting	C₂×C₄₇×#465 ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3056

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#465	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#15689633	C₂
hypercenter	C₂×#465	D₄₇
hypocenter	C₄₇	C₂×#3056
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#465

2898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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)(56,63)(59,68)(62,69)(64,67);(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)(51,53);(54,56,57,64,58,63,66,67)(55,59,60,62,61,68,65,69);(50,51)(52,53)(54,55,58,61)(56,62,63,69)(57,65,66,60)(59,67,68,64);(48,49)

Conjugacy classes & character table

1076 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036719);

C2×#20253011 #27083429