C₂×#20252920 #27083195

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

Aliases

direct product

C₂×#20252920

Splittings

Sylow

C₄₇ ⋊ C₂×#3044

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#137×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×#41×C₄₇	C₂²
hypercenter	C₂²×#137	D₄₇
hypocenter	C₄₇	C₂×#3044
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#137

2666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 85 points:

(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)(56,65)(60,74)(64,79)(66,81)(73,84)(75,82)(78,85)(80,83);(54,57,59,69)(55,61,63,76)(56,64,65,79)(58,71,72,68)(60,73,74,84)(62,77,70,67)(66,83,81,80)(75,85,82,78);(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)(50,51,52,53)(54,56)(55,60)(57,64)(58,66)(59,65)(61,73)(62,75)(63,74)(67,78)(68,80)(69,79)(70,82)(71,83)(72,81)(76,84)(77,85);(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)(51,53)(54,55,59,63)(56,60,65,74)(57,67,69,77)(58,70,72,62)(61,71,76,68)(64,78,79,85)(66,82,81,75)(73,83,84,80);(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, 1036485);

C2×#20252920 #27083195