C₂×#20272653 #27293217

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272653

Splittings

Sylow

C₅₃ ⋊ C₂×#1321

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#408	C₂
socle	C₂²×C₅₃	#323
prime radical	C₂²×#41×C₅₃	C₄
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	C₂×#408	D₅₃
hypocenter	C₅₃	C₂×#1321
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#408

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 87 points:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78693);

C2×#20272653 #27293217