C₂²×#15689138 #27079869

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#15689138

Splittings

Fitting	C₂²×#104×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#421

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#104×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×#41×D₄₇	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#104	D₄₇
hypocenter	C₄₇	C₂²×#421
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#104

5198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 3008

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

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033159);

C22×#15689138 #27079869