D₅₃×#1610 #27356306

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#1610

Splittings

Fitting	C₅₃×#1610 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1610

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1610	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#537×D₅₃	C₂
hypercenter	#1610	D₅₃
hypocenter	C₅₃	C₂×#1610
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1610

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 77 points:

(9,12,13,19)(10,15,16,22)(11,17,18,24)(14,20,21,23)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27);(2,5)(3,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(1,2)(3,5)(4,7)(6,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(26,27)(28,29)(30,31)(32,33)(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)

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, 141782);

D53×#1610 #27356306