C₂³×C₁₃×D₁₃ #72259278

order	2704
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂³×C₁₃×D₁₃

Splittings

Fitting, socle	C₂³×C₁₃² ⋊ C₂
derived	C₁₃ ⋊ C₂⁴×C₁₃
Sylow	C₁₃² ⋊ C₂⁴

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₁₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂⁴×C₁₃
Frattini	C₁	G
Fitting, socle	C₂³×C₁₃²	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂³×C₁₃

364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 4 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×D₁₃², C₂³×#11759817
maximal normal subgroup of	C₂³×D₁₃², C₂³×#11759817
maximal quotient of	C₁₃×#88251

GAP

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

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

C23×C13×D13 #72259278