D₅₃×#2161 #27537145

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2161

Splittings

Fitting	C₅₃×#2161 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2161

Subgroups

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2161	C₂
socle	C₂×C₅₃	C₂²×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2161	D₅₃
hypocenter	C₅₃	C₂×#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(3,14)(8,23)(12,26)(15,30)(16,17)(22,29)(24,27)(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);(1,3,13,29,6,17,20,30,7,14,19,22,5,16,4,15)(2,8,18,28,10,24,25,26,11,23,32,31,21,27,9,12);(1,2,5,10,7,11,6,21)(3,12,16,28,14,26,17,31)(4,18,19,25,20,32,13,9)(8,22,24,30,23,29,27,15);(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, 322621);

D53×#2161 #27537145