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Cayley diagram

C2×C3×Q8 #244

order48
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#1020⋊C2 (order 384) · Inn(G) = C22 · Out(G) = C22×S4
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×Q8

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C22
commutator, Frattini, cosocleC2C23×C3
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of#67460, #67463, #67542, …
Frattini subgroup of#10588154, #10588155, #10588219, …
Fitting subgroup of#835, #836, C2×#215, …
maximal subgroup of#835, #836, C3×#101, …
maximal normal subgroup of#835, #836, C3×#101, …
maximal quotient ofC2×C3×C4⋊C4, C3×C4×Q8, C3×C22⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 46);