#41×#10625354 #27123392

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

Aliases

direct product

#41×#10625354

Splittings

Fitting	Q₈×#41×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #41²

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	Q₈×#41×C₄₇	C₂
hypercenter	Q₈×#41	D₄₇
hypocenter	C₄₇	#41²
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ Q₈×#41

7812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 3008

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

Conjugacy classes & character table

1250 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076682);