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C23×C4×3−1+2 #11722683

order864
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C26

Aliases

direct productC23×C4×3−1+2

Subgroups

subgroupG/H
centerC23×C3×C4C32
commutatorC3C23×C32×C4
Frattini, cosocleC2×C3C24×C32
socleC24×C3C2×C32
prime radicalC24×C32C2×C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C23×C3×C4 ◁ G

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 288

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#63389⋊C3, #10588887⋊C3, C2×3−1+2⋊#95, …
maximal subgroup ofC2×#63389⋊C3, #10588887⋊C3, C2×3−1+2⋊#95, …
maximal normal subgroup ofC2×#63389⋊C3, #10588887⋊C3, C2×3−1+2⋊#95, …
maximal quotient ofC22×C42×3−1+2, C22×C22⋊C4×3−1+2, C22×C4⋊C4×3−1+2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 2650);