C₄×#16810810 #27390074

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₄×#16810810

Splittings

Fitting	C₄×H(5,𝔽₂)×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133186
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₄×H(5,𝔽₂)×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#16810810	C₂
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₄×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₄×B(4,𝔽₂)
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×H(5,𝔽₂)

1910 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 287 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 — 4 generators on 65 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)(58,60)(59,62)(61,64)(63,65);(58,59)(60,62)(61,63)(64,65);(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)(62,65);(54,55,56,57)

Conjugacy classes & character table

1832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 175550);