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Cayley diagram

C4×Dic24 #69064

order384
isoclinismisoclinic class of D48
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×Dic24

Splittings

SylowC3 ⋊ C4×Q32

Subgroups

subgroupG/H
centerC2×C4D24
commutatorC3×C8C22×C4
FrattiniC2×C8C22×D3
FittingC3×C4×C16C2
socle, prime radicalC22×C3C2×D8
cosocleC2×C3×C8C23
hypercenterC4×C16D3
hypocenterC3C4×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×C16

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(8,10,19,35,12,21,25,27,13,22,30,39,24,36,11,20)(9,14,23,33,16,28,31,34,17,29,38,37,26,32,15,18);(2,3)(8,9,13,17)(10,18,22,34)(11,23,25,38)(12,26,24,16)(14,27,29,20)(15,30,31,19)(21,37,36,33)(28,35,32,39);(4,5,6,7)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10682639, #10682641, #10682645, …
maximal normal subgroup of#10682639, #10682641, #10682645, …
maximal quotient ofC8×Dic24, #10707105, #10707179, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1640);