D₅₃×#2245 #27539407

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

D₅₃×#2245

Splittings

Fitting	C₅₃×#2245 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2245

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2245	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2245	D₅₃
hypocenter	C₅₃	C₂×#2245
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2245

582 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 points:

(2,6)(5,8)(9,12,13,27,29,39,23,26,16,28,30,25,15,24,14,11)(10,18,19,36,31,40,32,35,22,37,38,34,21,33,20,17)(41,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43);(1,3,4,7)(2,5,6,8)(9,11,23,39,15,25,13,12,16,26,14,24,29,27,30,28)(10,17,32,40,21,34,19,18,22,35,20,33,31,36,38,37);(1,2)(3,5)(4,6)(7,8)(9,10,16,22)(11,18,26,37)(12,17,28,35)(13,20,30,32)(14,19,23,38)(15,31,29,21)(24,36,39,34)(25,40,27,33);(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)(86,87)(88,89)(90,91)(92,93)

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, 324883);