D₅₃×#2132 #27536918

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

Aliases

direct product

D₅₃×#2132

Splittings

Fitting	C₅₃×#2132 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2132

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₅₃×#2132	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#532×D₅₃	C₂
cosocle	C₂×Q₈×C₅₃	C₂⁴
hypercenter	#2132	D₅₃
hypocenter	C₅₃	C₂×#2132
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2132

1306 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:

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

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, 322394);

D53×#2132 #27536918