C₈×#13133158 #27295812

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

Aliases

direct product

C₈×#13133158

Splittings

Fitting	C₈×D₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₈×D₁₆

Subgroups

	subgroup	G/H
center	C₂×C₈	#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×#13133158	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×D₈	D₅₃
hypocenter	C₅₃	C₈×D₁₆
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×D₈

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 points:

(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)(63,67)(64,71)(65,72)(68,69)(70,76)(73,77)(74,75);(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)(62,63)(64,67)(65,74)(66,70)(68,71)(69,72)(73,76)(75,77);(54,55,56,58,57,59,60,61)

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81288);