C₂×#20276689 #27538026

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

Aliases

direct product

C₂×#20276689

Splittings

Fitting	C₂×C₅₃×#361 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2194

Subgroups

	subgroup	G/H
center	C₂²	#16810758
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#361	C₂
socle	C₂²×C₅₃	#447
prime radical	C₂×#16811424	C₂
hypercenter	C₂×#361	D₅₃
hypocenter	C₅₃	C₂×#2194
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

1130 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)(56,58)(57,63)(59,68)(60,70)(61,71)(62,64)(65,82)(66,83)(67,77)(69,74)(72,80)(73,75)(76,78)(79,87)(81,85)(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)(59,73)(60,69)(61,71)(63,64)(65,79)(66,81)(67,72)(68,75)(70,74)(76,86)(77,80)(78,84)(82,87)(83,85);(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)(56,57,61,66)(58,67,71,82)(59,72,76,87)(60,77,74,65)(62,78,81,68)(63,70,83,69)(64,84,85,75)(73,79,86,80);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323502);

C2×#20276689 #27538026