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Cayley diagram

C23×C4⋊Q8 #62665

order256
nilpotency class2
isoclinismisoclinic class of #120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productC23×C4⋊Q8

Subgroups

subgroupG/H
center, socle, prime radicalC25C23
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C22 ▷ C1

6095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×C4⋊Q8, C4⋊Q8×AGL(1,𝔽8)
maximal subgroup of#154406, #154426, #154764, …
maximal normal subgroup of#154406, #154426, #154764, …
maximal quotient ofC23×#384, C23×#389, C23×#391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55653);