D₅₃×#1328 #27293281

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

Aliases

direct product

D₅₃×#1328

Splittings

Fitting	C₅₃×#1328 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1328

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1328	C₂
socle	C₂×C₅₃	C₂×#324
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1328	D₅₃
hypocenter	C₅₃	C₂×#1328
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1328

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 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)(54,56)(55,60,61,71)(57,65)(58,66)(59,64)(62,75,73,70)(63,72,74,82)(67,77)(68,79)(69,78)(76,84,83,81)(80,85);(54,55,57,62,58,63,68,76,59,61,67,73,69,74,80,83)(56,60,65,70,66,72,79,81,64,71,77,75,78,82,85,84);(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, 78757);