D₅₃×#2239 #27538979

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#2239

Splittings

Fitting	C₅₃×#2239 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2239

Subgroups

	subgroup	G/H
center	C₂	#481×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2239	C₂
socle	C₂×C₅₃	C₂×#481
prime radical	C₂³×D₅₃	SD₁₆
hypercenter	#2239	D₅₃
hypocenter	C₅₃	C₂×#2239
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #2239

552 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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324455);

D53×#2239 #27538979