C₂×#20272668 #27293283

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

Aliases

direct product

C₂×#20272668

Splittings

Sylow

C₅₃ ⋊ C₂×#1328

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133132
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#424	C₂
socle	C₂²×C₅₃	#324
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	C₂×#424	D₅₃
hypocenter	C₅₃	C₂×#1328
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#424

378 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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)(57,62,63,73)(59,67)(60,68)(61,66)(64,77,75,72)(65,74,76,84)(69,79)(70,81)(71,80)(78,86,85,83)(82,87);(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)(56,57,59,64,60,65,70,78,61,63,69,75,71,76,82,85)(58,62,67,72,68,74,81,83,66,73,79,77,80,84,87,86);(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, 78759);