C₂×#20272703 #27294237

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272703

Splittings

Sylow

C₅₃ ⋊ C₂×#1340

Subgroups

	subgroup	G/H
center	C₂²	#16810357
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#470	C₂
socle	C₂²×C₅₃	#327
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	C₂×#470	D₅₃
hypocenter	C₅₃	C₂×#1340
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#470

444 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)(56,58,62,69)(57,63,66,79)(59,73,74,85)(60,71,61,70)(64,80,81,87)(65,82,77,78)(67,83,84,72)(68,75,86,76);(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)(58,67,69,84)(59,72,74,83)(60,65)(61,77)(62,66)(63,68)(64,75,81,76)(70,80,71,87)(73,82)(78,85)(79,86);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79713);

C2×#20272703 #27294237