D₅₃×#2201 #27538132

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2201

Splittings

Fitting	C₅₃×#2201 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2201

Subgroups

	subgroup	G/H
center	C₂	B(4,𝔽₂)×D₅₃
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2201	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
prime radical	H(5,𝔽₂)×D₅₃	C₂²
hypercenter	#2201	D₅₃
hypocenter	C₅₃	C₂×#2201
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2201

1136 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 139 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 — 4 generators on 85 points:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323608);