C₂×SD₆₄×D₅₃ #27542209

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

Aliases

direct product

C₂×SD₆₄×D₅₃

Splittings

Fitting	C₂×C₅₃×SD₆₄ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×SD₆₄

Subgroups

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×SD₆₄	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂×D₁₆×D₅₃	C₂
hypercenter	C₂×SD₆₄	D₅₃
hypocenter	C₅₃	C₂²×SD₆₄
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×SD₆₄

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

normal subgroups only

This group has 502 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

(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);(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)(57,62)(58,67)(59,68)(60,72)(63,64)(65,83)(66,77)(69,86)(70,78)(71,75)(73,81)(74,76)(79,87)(80,82)(84,85);(56,57,61,66)(58,62,70,77)(59,71,73,87)(60,74,72,65)(63,78,80,67)(64,81,82,68)(69,83,84,76)(75,86,79,85);(54,55)

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327685);