D₅₃×#2169 #27537252

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2169

Splittings

Fitting	C₅₃×#2169 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2169

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2169	C₂
prime radical	C₂×D₄×D₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2169	D₅₃
hypocenter	C₅₃	C₂×#2169
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2169

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 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)(58,60,62,69,63,64,67,72,61,68,70,73,71,59,65,66);(55,57)(59,64)(60,68)(66,72)(69,73);(54,55)(56,57)(58,59,62,66,63,60,67,69,61,64,70,72,71,68,65,73);(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

1568 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322728);

D53×#2169 #27537252