C₂×#20272878 #27304397

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

Aliases

direct product

C₂×#20272878

Splittings

Sylow

C₅₃ ⋊ C₂×#1408

Subgroups

	subgroup	G/H
center	C₂²	#16810416
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	#127	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂²×C₅₃	#353
cosocle	#127×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	C₂×#1408
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#127 ◁ C₂×#493

526 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 71 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)(57,61)(59,68)(60,67)(62,70)(63,65)(64,71)(66,69);(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)(58,65,66,61)(59,63,69,62)(60,64)(67,70,68,71);(54,55)

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89873);