C₂×#20257528 #27123171

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

Aliases

direct product

C₂×#20257528

Splittings

Fitting	C₂×C₄₇×H(3,𝔽₄) ⋊ C₂
Sylow	C₄₇ ⋊ C₂×#3529

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×H(3,𝔽₄)	C₂
socle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₄₇
hypocenter	C₄₇	C₂×#3529
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

7418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3008

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076461);

C2×#20257528 #27123171