D₅₃×#1422 #27304580

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

D₅₃×#1422

Splittings

Fitting	C₅₃×#1422 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1422

Subgroups

	subgroup	G/H
center	C₁₆	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1422	C₂
socle	C₂×C₅₃	C₂×C₄⋊C₁₆
prime radical	C₂²×D₅₃	M₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1422	D₅₃
hypocenter	C₅₃	C₂×#1422
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

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

normal subgroups only

This group has 214 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:

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

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90056);