D₅₃×#1321 #27293215

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

Aliases

direct product

D₅₃×#1321

Splittings

Fitting	C₅₃×#1321 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1321

Subgroups

	subgroup	G/H
center	C₄	#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1321	C₂
socle	C₂×C₅₃	C₂×#323
prime radical	C₂×#41×D₅₃	C₄
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1321	D₅₃
hypocenter	C₅₃	C₂×#1321
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1321

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

normal subgroups only

This group has 518 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:

(1,3)(2,8)(4,17)(5,19)(6,13)(7,15)(9,25)(10,23)(11,24)(12,16)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,5,10,6,9,20,26,7,11,21,28,4,16,18,27)(3,12,14,29,13,8,22,23,15,25,19,31,17,24,30,32);(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

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, 78691);