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## G = C32⋊D8order 144 = 24·32

### The semidirect product of C32 and D8 acting via D8/C2=D4

Aliases: C32⋊D8, C2.3S3≀C2, (C3×C6).3D4, D6⋊S31C2, C322C81C2, C3⋊Dic3.1C22, SmallGroup(144,117)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C32⋊D8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C32⋊D8
 Upper central C1 — C2

Generators and relations for C32⋊D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

12C2
12C2
2C3
2C3
6C22
6C22
9C4
2C6
2C6
4S3
4S3
12C6
12C6
9C8
9D4
9D4
2D6
2D6
6Dic3
6Dic3
9D8

Character table of C32⋊D8

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 8A 8B size 1 1 12 12 4 4 18 4 4 12 12 12 12 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 √2 -√2 orthogonal lifted from D8 ρ7 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 -√2 √2 orthogonal lifted from D8 ρ8 4 4 0 -2 1 -2 0 -2 1 0 0 1 1 0 0 orthogonal lifted from S3≀C2 ρ9 4 4 2 0 -2 1 0 1 -2 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ10 4 4 -2 0 -2 1 0 1 -2 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ11 4 4 0 2 1 -2 0 -2 1 0 0 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ12 4 -4 0 0 -2 1 0 -1 2 -√-3 √-3 0 0 0 0 complex faithful ρ13 4 -4 0 0 -2 1 0 -1 2 √-3 -√-3 0 0 0 0 complex faithful ρ14 4 -4 0 0 1 -2 0 2 -1 0 0 -√-3 √-3 0 0 complex faithful ρ15 4 -4 0 0 1 -2 0 2 -1 0 0 √-3 -√-3 0 0 complex faithful

Permutation representations of C32⋊D8
On 24 points - transitive group 24T218
Generators in S24
(1 21 12)(2 22 13)(3 14 23)(4 15 24)(5 17 16)(6 18 9)(7 10 19)(8 11 20)
(1 21 12)(2 13 22)(3 14 23)(4 24 15)(5 17 16)(6 9 18)(7 10 19)(8 20 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 24)(18 23)(19 22)(20 21)

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

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

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

G:=TransitiveGroup(24,218);

On 24 points - transitive group 24T219
Generators in S24
(1 21 11)(2 12 22)(3 13 23)(4 24 14)(5 17 15)(6 16 18)(7 9 19)(8 20 10)
(1 11 21)(2 12 22)(3 23 13)(4 24 14)(5 15 17)(6 16 18)(7 19 9)(8 20 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)

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

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

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

G:=TransitiveGroup(24,219);

C32⋊D8 is a maximal subgroup of
C32⋊D85C2  C32⋊D8⋊C2  C3⋊S3⋊D8  C62.12D4  C62.13D4  C33⋊D8  C322D24
C32⋊D8 is a maximal quotient of
C32⋊D16  C32⋊SD32  C32⋊Q32  C62.3D4  C62.7D4  He3⋊D8  C33⋊D8  C322D24

Matrix representation of C32⋊D8 in GL4(𝔽7) generated by

 3 2 4 3 4 5 5 6 3 3 6 1 0 0 0 1
,
 6 2 1 1 2 6 6 1 0 0 1 0 0 0 0 2
,
 0 3 2 1 1 1 5 4 4 3 5 3 3 3 4 1
,
 0 1 1 0 1 0 1 0 0 0 6 0 0 0 0 1
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[0,1,4,3,3,1,3,3,2,5,5,4,1,4,3,1],[0,1,0,0,1,0,0,0,1,1,6,0,0,0,0,1] >;

C32⋊D8 in GAP, Magma, Sage, TeX

C_3^2\rtimes D_8
% in TeX

G:=Group("C3^2:D8");
// GroupNames label

G:=SmallGroup(144,117);
// by ID

G=gap.SmallGroup(144,117);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,73,218,116,50,964,730,256,299,881]);
// Polycyclic

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

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