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G = C3×D8order 48 = 24·3

Direct product of C3 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×D8, D4⋊C6, C81C6, C243C2, C6.14D4, C12.17C22, (C3×D4)⋊4C2, C4.1(C2×C6), C2.3(C3×D4), SmallGroup(48,25)

Series: Derived Chief Lower central Upper central

C1C4 — C3×D8
C1C2C4C12C3×D4 — C3×D8
C1C2C4 — C3×D8
C1C6C12 — C3×D8

Generators and relations for C3×D8
 G = < a,b,c | a3=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C6
4C6
2C2×C6
2C2×C6

Character table of C3×D8

 class 12A2B2C3A3B46A6B6C6D6E6F8A8B12A12B24A24B24C24D
 size 114411211444422222222
ρ1111111111111111111111    trivial
ρ2111-111111-1-111-1-111-1-1-1-1    linear of order 2
ρ311-111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ411-1-111111-1-1-1-111111111    linear of order 2
ρ511-1-1ζ3ζ321ζ32ζ3ζ6ζ65ζ6ζ6511ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 6
ρ611-11ζ3ζ321ζ32ζ3ζ32ζ3ζ6ζ65-1-1ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ7111-1ζ32ζ31ζ3ζ32ζ65ζ6ζ3ζ32-1-1ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ81111ζ3ζ321ζ32ζ3ζ32ζ3ζ32ζ311ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ91111ζ32ζ31ζ3ζ32ζ3ζ32ζ3ζ3211ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ1011-11ζ32ζ31ζ3ζ32ζ3ζ32ζ65ζ6-1-1ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ1111-1-1ζ32ζ31ζ3ζ32ζ65ζ6ζ65ζ611ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 6
ρ12111-1ζ3ζ321ζ32ζ3ζ6ζ65ζ32ζ3-1-1ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ13220022-222000000-2-20000    orthogonal lifted from D4
ρ142-200220-2-20000-220022-2-2    orthogonal lifted from D8
ρ152-200220-2-200002-200-2-222    orthogonal lifted from D8
ρ162200-1--3-1+-3-2-1+-3-1--30000001--31+-30000    complex lifted from C3×D4
ρ172200-1+-3-1--3-2-1--3-1+-30000001+-31--30000    complex lifted from C3×D4
ρ182-200-1+-3-1--301+-31--30000-220083ζ38ζ383ζ328ζ3287ζ385ζ387ζ3285ζ32    complex faithful
ρ192-200-1--3-1+-301--31+-300002-20087ζ3285ζ3287ζ385ζ383ζ328ζ3283ζ38ζ3    complex faithful
ρ202-200-1+-3-1--301+-31--300002-20087ζ385ζ387ζ3285ζ3283ζ38ζ383ζ328ζ32    complex faithful
ρ212-200-1--3-1+-301--31+-30000-220083ζ328ζ3283ζ38ζ387ζ3285ζ3287ζ385ζ3    complex faithful

Permutation representations of C3×D8
On 24 points - transitive group 24T40
Generators in S24
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 17)(8 9 18)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)

G:=sub<Sym(24)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)>;

G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,17)(8,9,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24) );

G=PermutationGroup([(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,17),(8,9,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)])

G:=TransitiveGroup(24,40);

Matrix representation of C3×D8 in GL2(𝔽7) generated by

40
04
,
06
13
,
31
64
G:=sub<GL(2,GF(7))| [4,0,0,4],[0,1,6,3],[3,6,1,4] >;

C3×D8 in GAP, Magma, Sage, TeX

C_3\times D_8
% in TeX

G:=Group("C3xD8");
// GroupNames label

G:=SmallGroup(48,25);
// by ID

G=gap.SmallGroup(48,25);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,141,723,368,58]);
// Polycyclic

G:=Group<a,b,c|a^3=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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