C₄×#16810411 #27299227

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×#16810411

Splittings

Fitting	C₄×#100×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×AΣL(1,𝔽₁₆)

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₄×#100×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#13133185	C₂²
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₄×#100	D₅₃
hypocenter	C₅₃	C₄×AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ C₄×#100

844 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 65
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 65 points:

(1,2)(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)(59,62)(63,64);(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)(58,59)(60,64,65,62)(61,63);(54,55,56,57)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84703);