D₅₃×#2030 #27411490

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2030

Splittings

Fitting	C₅₃×#2030 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2030

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2030	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂⁴×D₅₃	C₂³
hypercenter	#2030	D₅₃
hypocenter	C₅₃	C₂×#2030
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2030

1482 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 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)(62,65,66,72)(63,68,69,76)(64,70,71,77)(67,73,74,75);(54,56,57,61)(55,58,59,60)(62,64)(63,67)(65,70)(66,71)(68,75)(69,74)(72,77)(73,76);(54,55,57,59)(56,60,61,58)(62,63)(64,67)(65,68)(66,69)(70,75)(71,74)(72,76)(73,77);(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, 196966);