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3−1+2×#3498 #72244709

order3456
nilpotency class2
automorphism?
homologyAbelianization (H1) = C25×C32 · Schur multiplier (H2) = C29

Aliases

direct product3−1+2×#3498

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC22×C3C25×C32
prime radicalC32×#3498C3
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

6992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup ofAGL(1,ℤ/9)×#3498
maximal subgroup ofAGL(1,ℤ/9)×#3498
maximal normal subgroup ofAGL(1,ℤ/9)×#3498

GAP

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

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