D₅₃×#2129 #27536882

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

Aliases

direct product

D₅₃×#2129

Splittings

Fitting	C₅₃×#2129 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2129

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2129	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2129	D₅₃
hypocenter	C₅₃	C₂×#2129
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2129

1230 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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322358);

D53×#2129 #27536882