D₅₃×#1350 #27294339

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

Aliases

direct product

D₅₃×#1350

Splittings

Fitting	C₅₃×#1350 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1350

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1350	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1350	D₅₃
hypocenter	C₅₃	C₂×#1350
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1350

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

normal subgroups only

This group has 446 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,39)(13,24,14,23)(17,34,36,37)(18,32,19,31)(21,26,38,35)(22,28,40,29)(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);(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,20,14,19)(11,21,23,34,25,38,24,37)(12,26,28,17,30,35,29,36)(16,27,31,22,33,39,32,40);(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

812 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79815);