C₂²×#15689786 #27123191

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×#15689786

Splittings

Fitting	C₂²×C₂²⋊Q₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×D₄×Q₈

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₂²⋊Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂³×C₂²⋊D₄₇	C₂²
hypercenter	C₂²×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈

11798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 6 generators on 63 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)(56,59,60,57)(58,61,63,62);(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)(53,55);(56,58,60,63)(57,61,59,62);(52,53)(54,55)(56,57,60,59)(58,62,63,61);(50,51);(48,49)

Conjugacy classes & character table

1388 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076481);

C22×#15689786 #27123191