C₂²×#15689081 #27079672

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#15689081

Splittings

Sylow

C₄₇ ⋊ C₂²×#405

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	C₂²×#405
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

3518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 71 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)(58,64)(61,67)(65,70)(68,71);(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)(52,53,54,55)(56,58)(57,61)(59,65)(60,64)(62,68)(63,67)(66,70)(69,71);(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)(56,57,59,62,60,63,66,69)(58,61,65,68,64,67,70,71);(50,51);(48,49)

Conjugacy classes & character table

2000 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1032962);

C22×#15689081 #27079672