C₂×#20256793 #27123122

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#20256793

Splittings

Sylow

C₄₇ ⋊ C₂×#3458

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	C₂×#3458
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

7022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 6 generators on 73 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)(58,61,62,69)(59,65,64,71)(60,68,67,73)(63,70,66,72);(59,64)(63,66)(65,71)(70,72);(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)(50,52,53,57)(51,54,55,56)(60,67)(63,66)(68,73)(70,72);(58,60)(59,63)(61,68)(62,67)(64,66)(65,70)(69,73)(71,72);(50,51,53,55)(52,56,57,54)(58,59)(60,66)(61,65)(62,64)(63,67)(68,72)(69,71)(70,73);(48,49)

Conjugacy classes & character table

1928 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076412);

C2×#20256793 #27123122