C₂²×#15689872 #27123159

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

Aliases

direct product

C₂²×#15689872

Splittings

Fitting	C₂²×C₄⋊Q₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#563

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂³×D₁₈₈	C₂²
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#563
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

7238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 6 generators on 75 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)(53,57)(56,59)(61,67)(62,70)(66,75)(69,71);(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)(53,57)(54,58)(60,63,64,72)(61,66,67,75)(62,69,70,71)(65,73,68,74);(52,54)(53,56)(55,58)(57,59)(60,62,64,70)(61,65,67,68)(63,71,72,69)(66,74,75,73);(52,53)(54,56)(55,57)(58,59)(60,61,64,67)(62,68,70,65)(63,66,72,75)(69,74,71,73);(50,51);(48,49)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076449);

C22×#15689872 #27123159