C₂×#20276552 #27537320

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20276552

Splittings

Sylow

C₅₃ ⋊ C₂×#2170

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	C₂×#2170
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

662 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:

(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)(47,48)(49,50)(51,52)(56,59,61,68,60,73,75,79,62,74,77,81,76,87,80,63)(57,58,65,70,66,71,83,85,67,72,84,86,64,69,82,78);(56,58)(57,63)(59,65)(60,71)(61,70)(62,72)(64,81)(66,68)(67,79)(69,76)(73,83)(74,84)(75,85)(77,86)(78,80)(82,87);(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)(48,49)(50,51)(52,53)(56,57,61,65,60,66,75,83,62,67,77,84,76,64,80,82)(58,68,70,73,71,79,85,74,72,81,86,87,69,63,78,59);(54,55)

Conjugacy classes & character table

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322796);

C2×#20276552 #27537320