C₂×#20272902 #27304551

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

Aliases

direct product

C₂×#20272902

Splittings

Sylow

C₅₃ ⋊ C₂×#1418

Subgroups

	subgroup	G/H
center	C₂²	#16810424
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	#357
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	C₂×#1418
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

302 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)(56,57)(58,67)(59,72)(60,75)(61,63)(62,76)(64,71)(65,73)(66,78)(68,81)(69,82)(70,79)(74,77)(80,85)(83,87)(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)(57,62,63,78)(58,66)(59,71)(60,73)(64,82,81,86)(65,83,75,80)(67,72,79,77)(68,74)(69,87)(70,76)(84,85);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90027);

C2×#20272902 #27304551