D₅₃×#1369 #27295652

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

Aliases

direct product

D₅₃×#1369

Splittings

Fitting	C₅₃×#1369 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1369

Subgroups

	subgroup	G/H
center	C₄	#106×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1369	C₂
socle	C₂×C₅₃	C₂×#334
prime radical	C₂²×D₅₃	#108
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1369	D₅₃
hypocenter	C₅₃	C₂×#1369
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

222 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(1,3,4,15,6,17,21,16,8,19,22,30,7,18,5,23)(2,9,10,14,12,25,20,29,13,26,11,27,24,32,28,31)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,5,11,7,12,22,28,8,13,21,10,6,24,4,20)(3,14,16,25,18,29,15,26,19,27,23,32,17,31,30,9);(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81128);