C₂×(C₃³⋊C₁₃)⋊C₃ #72224468

order	2106
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×(C₃³⋊C₁₃)⋊C₃

Splittings

Fitting, socle	C₂×C₃³ ⋊ C₁₃⋊C₃
derived	C₃³⋊C₁₃ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₃³⋊C₁₃)⋊C₃
commutator, cosocle, hypocenter	C₃³⋊C₁₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₁₃⋊C₃
LCS	G ▷ C₃³⋊C₁₃
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

ΣL(2,𝔽₂₇)

GAP

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

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

C2×(C33⋊C13)⋊C3 #72224468