D₅₃×#2212 #27538460

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2212

Splittings

Fitting	C₅₃×#2212 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2212

Subgroups

	subgroup	G/H
center	C₂	#459×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2212	C₂
socle	C₂×C₅₃	C₂×#459
prime radical	#509×D₅₃	C₂
hypercenter	#2212	D₅₃
hypocenter	C₅₃	C₂×#2212
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2212

706 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)(2,9)(3,13)(5,17)(6,23)(7,19)(8,24)(10,25)(11,28)(12,27)(14,30)(15,31)(16,22)(18,21)(20,26)(29,32)(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);(1,3,5,14,7,15,21,6)(2,8,10,11,12,22,20,29)(4,13,17,30,19,31,18,23)(9,24,25,28,27,16,26,32);(1,2)(3,11)(4,16)(5,20)(6,22)(7,12)(8,14)(9,13)(10,21)(15,29)(17,28)(18,32)(19,24)(23,25)(26,30)(27,31);(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, 323936);

D53×#2212 #27538460