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Cayley diagram

C2×C3×C8 #221

order48
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×D4 (order 32) · Inn(G) = C1 · Out(G) = C22×D4
holomorphG ⋊ Aut(G) = D3×#23882
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC2×C3×C8

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC4C22×C3
socle, prime radicalC22×C3C4
LCSG ▷ C1
UCSC1 ◁ G

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2,3,4,5,6,7,8)(11,12,13);(9,10)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

center ofC3×#346, C3×C22⋊C16, C3×C4⋊C16, …
derived subgroup of#67467, #67468, #67473, …
Frattini subgroup ofC4×C9⋊C16, #10588138, C4⋊(C9⋊C16), …
Fitting subgroup ofC2×C3⋊C16, #812, C8×Dic3, …
maximal subgroup ofC2×C3⋊C16, #812, C8×Dic3, …
maximal normal subgroup ofC2×C3⋊C16, #812, C8×Dic3, …
maximal quotient ofC3×C4×C8, C3×C22⋊C8, C3×C4⋊C8, …
outer automorphism group ofC7⋊C32, C7×D17, C9⋊C32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 23);