C₈×#13133163 #27295824

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

Aliases

direct product

C₈×#13133163

Splittings

Sylow

C₅₃ ⋊ C₈×Q₃₂

Subgroups

	subgroup	G/H
center	C₂×C₈	#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	C₈×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×Q₁₆

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

normal subgroups only

This group has 226 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) = 93
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 93 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)(62,64,67,75)(63,68,71,85)(65,77,80,93)(66,82,78,74)(69,87,88,92)(70,89,81,84)(72,90,91,76)(73,86,83,79);(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)(62,63,67,71)(64,72,75,91)(65,76,80,90)(66,81,78,70)(68,83,85,73)(69,86,88,79)(74,92,82,87)(77,89,93,84);(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, 81300);