C₂×#20276737 #27538251

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276737

Splittings

Fitting	C₂×C₅₃×#356 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2203

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂×Q₈×C₅₃	C₂⁴
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#356	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₂×#16810798	C₂
hypercenter	C₂×#356	D₅₃
hypocenter	C₅₃	C₂×#2203
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×#356

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

(4,10)(7,23)(9,25)(11,30)(12,13)(14,31)(16,22)(18,28)(20,24)(26,27)(29,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86);(3,5,8,17)(4,9,13,27)(6,19,21,15)(7,22,23,16)(10,25,12,26)(11,29,18,14)(20,33,24,34)(28,31,30,32);(3,4)(5,14)(6,18)(7,10)(8,13)(9,15)(11,21)(12,23)(16,31)(17,29)(19,27)(20,28)(22,32)(24,30)(25,33)(26,34)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323727);

C2×#20276737 #27538251