D₅₃×#2120 #27536809

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

Aliases

direct product

D₅₃×#2120

Splittings

Fitting	C₅₃×#2120 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2120

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2120	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#525×D₅₃	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2120	D₅₃
hypocenter	C₅₃	C₂×#2120
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2120

1398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 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)(54,56,58,63)(55,60,61,67)(57,62,65,69)(59,66,64,68);(55,59)(60,66)(61,64)(67,68);(54,55)(56,60)(57,64,65,59)(58,61)(62,68,69,66)(63,67);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322285);

D53×#2120 #27536809