D₅₃×#1658 #27365275

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#1658

Splittings

Fitting	C₅₃×#1658 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1658

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1658	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#127×D₅₃	C₂²
hypercenter	#1658	D₅₃
hypocenter	C₅₃	C₂×#1658
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1658

882 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 3392

A minimal faithful permutation representation — 4 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)(54,56,58,63)(55,60,61,69)(57,66,64,62)(59,67,65,68)(70,72)(71,75)(73,82)(74,79)(76,85)(77,84)(78,80)(81,83);(54,55,58,61)(56,60,63,69)(57,65,64,59)(62,67,66,68)(70,71,74,77)(72,75,79,84)(73,81,80,76)(78,85,82,83);(55,59)(57,64)(60,68)(61,65)(62,66)(67,69)(72,78)(73,80)(75,83)(76,81)(79,82)(84,85);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150751);