D₅₃×#2597 #27554669

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₅₃×#2597

Splittings

Fitting	C₅₃×#2597 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2597

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2597	C₂
prime radical	C₂²×D₄×D₅₃	C₂²
hypercenter	#2597	D₅₃
hypocenter	C₅₃	C₂×#2597
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2597

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 77 points:

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

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 340145);

D53×#2597 #27554669