Finite Group Databasebeta

← Prev (#67270) Next (#67272) →

Cayley diagram

(C2×D23)⋊C4 #67271

order368
isoclinismisoclinic class of Dic46
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect product(C2×D23)⋊C4

Splittings

FittingC2×C4×C23 ⋊ C2
SylowC23 ⋊ C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×D23
commutatorC2×C23C2×C4
FittingC2×C4×C23C2
socle, cosocleC22×C23C22
prime radicalC22×D23C2
hypercenterC2×C4D23
hypocenterC23C22⋊C4
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C2×C4

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 184

A minimal faithful permutation representation — 2 generators on 31 points:

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

Polycyclic presentation:

2^4,23|b^2=c,b^a=bd,e^a=e^22

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10625043, C4×D92, #10625046, …
maximal normal subgroup of#10625043, C4×D92, #10625046, …
maximal quotient of#10624987, #10624988, #10624991, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 13);