D₅₃×#2143 #27537017

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2143

Splittings

Fitting	C₅₃×#2143 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2143

Subgroups

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2143	C₂
socle	C₂×C₅₃	C₂²×#102
prime radical	C₂×#135×D₅₃	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2143	D₅₃
hypocenter	C₅₃	C₂×#2143
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2143

1154 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:

(3,13)(8,23)(12,28)(14,30)(15,22)(17,32)(24,25)(26,29)(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)(2,8)(4,17)(5,15)(6,22)(7,13)(9,26)(10,24)(11,23)(12,16)(14,19)(18,30)(20,32)(21,25)(27,29)(28,31);(1,2,5,10,7,11,6,21)(3,12,15,29,13,28,22,26)(4,16,18,27,20,31,19,9)(8,17,24,30,23,32,25,14);(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, 322493);

D53×#2143 #27537017