C₂×#20255964 #27094973

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#20255964

Splittings

Fitting	C₂×C₄×Q₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3352

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#3352
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 73 points:

(1,2)(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)(52,57)(55,56)(59,65)(63,71)(64,66)(70,72);(50,53,54,51)(52,56,57,55)(59,64)(60,67)(61,69)(63,70)(65,66)(71,72);(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)(50,52)(51,55)(53,56)(54,57)(58,60)(59,63)(61,68)(62,67)(64,72)(65,71)(66,70)(69,73);(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)(50,51,54,53)(52,56,57,55)(58,59,62,65)(60,63,67,71)(61,64,69,66)(68,72,73,70);(48,49)

Conjugacy classes & character table

1346 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1048263);

C2×#20255964 #27094973