C₂×#518×D₅₃ #28290669

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#518×D₅₃

Splittings

Fitting	C₂×C₅₃×#518 ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#518

Subgroups

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#518	C₂
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂³×D₄×D₅₃	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂×#518	D₅₃
hypocenter	C₅₃	C₂²×#518
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#518

14590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 6 generators on 67 points:

(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);(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)(61,65)(64,66);(62,67)(64,66);(60,62)(61,64)(63,67)(65,66);(56,57,58,59)(60,61)(62,66)(63,65)(64,67);(54,55)

Conjugacy classes & character table

1904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1076145);

C2×#518×D53 #28290669