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3−1+2×#92265 #72243256

order13122
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C34

Aliases

direct product3−1+2×#92265

Splittings

Fitting, Sylow3−1+2×#6900 ⋊ C2

Subgroups

subgroupG/H
centerC32C33×AGL(1,ℤ/9)
commutator, cosocleC32×C9C2×C34
Frattini, socleC33C34×D3
Fitting3−1+2×#6900C2
prime radicalC33×AGL(1,ℤ/9)C32
hypercenterC9×3−1+2AGL(1,ℤ/9)
hypocenterC9C2×3−1+22
LCSG ▷ C32×C9 ▷ C9
UCSC1 ◁ C32 ◁ C9×3−1+2

6934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of3−1+2×#92265⋊C2, AGL(1,ℤ/9)×#92265, #72243365
maximal normal subgroup of3−1+2×#92265⋊C2, AGL(1,ℤ/9)×#92265, #72243365

GAP

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

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