C₂×#20276403 #27536926

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276403

Splittings

Fitting	C₂×C₅₃×#356 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2133

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#356	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×#11719949	C₄
cosocle	C₂×Q₈×C₅₃	C₂⁴
hypercenter	C₂×#356	D₅₃
hypocenter	C₅₃	C₂×#2133
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×#356

1198 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)(58,67)(62,78)(66,80)(68,71)(69,74)(77,85)(79,87)(84,86);(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)(56,58)(57,62)(59,71)(60,74)(61,67)(63,80)(64,79)(65,78)(66,70)(68,73)(69,76)(72,77)(75,85)(81,86)(82,84)(83,87);(56,57,60,64,61,65,76,83)(58,66,69,84,67,80,74,86)(59,70,72,82,73,63,75,81)(62,77,79,71,78,85,87,68);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322402);

C2×#20276403 #27536926