C₂×H(5,𝔽₂)×C₁₃⋊C₃ #72276212

order	2496
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×H(5,𝔽₂)×C₁₃⋊C₃

Splittings

Fitting, Hall	C₂×C₁₃×H(5,𝔽₂) ⋊ C₃
Sylow	C₁₃ ⋊ C₂×C₃×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×C₁₃⋊C₃
commutator, cosocle	C₂×C₁₃	C₂⁵×C₃
Frattini	C₂	C₂⁵×C₁₃⋊C₃
Fitting	C₂×C₁₃×H(5,𝔽₂)	C₃
socle	C₂²×C₁₃	C₂⁴×C₃
hypercenter	C₂×H(5,𝔽₂)	C₁₃⋊C₃
hypocenter	C₁₃	C₂×C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

1796 conjugacy classes of subgroups; only the 1131 normal subgroups are shown.

normal subgroups only

This group has 1796 conjugacy classes of subgroups — too many to draw. Only its 1131 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) = 23
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 5 generators on 23 points:

(1,2)(12,13,15)(14,17,21)(16,19,18)(20,22,23);(4,8)(7,10)(11,12,14,15,18,17,21,23,22,13,16,20,19);(3,5,6,9)(4,7,8,10);(3,4)(5,7)(6,8)(9,10);(5,9)(7,10)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(12,13,15)(14,17,21)(16,19,18)(20,22,23), (4,8)(7,10)(11,12,14,15,18,17,21,23,22,13,16,20,19), (3,5,6,9)(4,7,8,10), (3,4)(5,7)(6,8)(9,10), (5,9)(7,10));