D₅₃×#1373 #27295683

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

Aliases

direct product

D₅₃×#1373

Splittings

Fitting	C₅₃×#1373 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1373

Subgroups

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1373	C₂
socle	C₂×C₅₃	C₂×#335
prime radical	C₂²×D₅₃	#108
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1373	D₅₃
hypocenter	C₅₃	C₂×#1373
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

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

normal subgroups only

This group has 230 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:

(1,3,15,30,6,18,22,32,7,19,5,17,21,31,4,16)(2,8,20,28,11,24,10,23,13,26,27,29,12,25,9,14)(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);(1,2,5,10,6,12,4,20,7,13,15,9,21,11,22,27)(3,14,17,24,18,29,16,8,19,23,30,25,31,28,32,26);(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

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81159);