D₅₃×#2133 #27536923

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

Aliases

direct product

D₅₃×#2133

Splittings

Fitting	C₅₃×#2133 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2133

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₅₃×#2133	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	C₂×#41×D₅₃	C₄
cosocle	C₂×Q₈×C₅₃	C₂⁴
hypercenter	#2133	D₅₃
hypocenter	C₅₃	C₂×#2133
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #2133

1198 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:

(3,12)(7,23)(11,25)(13,16)(14,19)(22,30)(24,32)(29,31)(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)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,18)(14,21)(17,22)(20,30)(26,31)(27,29)(28,32);(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,17,27,18,8,20,26)(7,22,24,16,23,30,32,13);(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, 322399);

D53×#2133 #27536923