C₂×#20276780 #27538351

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20276780

Splittings

Fitting	C₂×C₅₃×#358 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2211

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#358	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	C₂×#358	D₅₃
hypocenter	C₅₃	C₂×#2211
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 91 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)(60,62,63,71)(61,67,66,81)(64,77,74,91)(65,80,75,73)(68,84,82,90)(69,85,79,83)(70,87,86,76)(72,89,88,78);(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)(56,57,58,59)(61,66)(67,81)(68,82)(69,79)(70,86)(76,87)(83,85)(84,90);(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)(57,59)(60,61)(62,70)(63,66)(64,76)(65,79)(67,72)(68,78)(69,75)(71,86)(73,90)(74,87)(77,85)(80,84)(81,88)(82,89)(83,91);(54,55)

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323827);

C2×#20276780 #27538351