D₅₃×#1318 #27293189

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

D₅₃×#1318

Splittings

Fitting	C₅₃×#1318 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1318

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1318	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂²×D₄×D₅₃	C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	#1318	D₅₃
hypocenter	C₅₃	C₂×#1318
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1318

834 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 101 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) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 69 points:

(10,13)(11,16)(14,15)(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);(1,2,3,5,4,6,7,8)(9,10)(11,14,16,13)(12,15);(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, 78665);