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G = C32⋊D8order 144 = 24·32

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

non-abelian, soluble, monomial

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

C1C32C3⋊Dic3 — C32⋊D8
C1C32C3×C6C3⋊Dic3D6⋊S3 — C32⋊D8
C32C3×C6C3⋊Dic3 — C32⋊D8
C1C2

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
6C2×C6
6Dic3
6C2×C6
4C3×S3
4C3×S3
9D8
6C3⋊D4
6C3⋊D4
2S3×C6
2S3×C6

Character table of C32⋊D8

 class 12A2B2C3A3B46A6B6C6D6E6F8A8B
 size 111212441844121212121818
ρ1111111111111111    trivial
ρ211-1-111111-1-1-1-111    linear of order 2
ρ311-1111111-1-111-1-1    linear of order 2
ρ4111-11111111-1-1-1-1    linear of order 2
ρ5220022-222000000    orthogonal lifted from D4
ρ62-200220-2-200002-2    orthogonal lifted from D8
ρ72-200220-2-20000-22    orthogonal lifted from D8
ρ8440-21-20-21001100    orthogonal lifted from S3≀C2
ρ94420-2101-2-1-10000    orthogonal lifted from S3≀C2
ρ1044-20-2101-2110000    orthogonal lifted from S3≀C2
ρ1144021-20-2100-1-100    orthogonal lifted from S3≀C2
ρ124-400-210-12--3-30000    complex faithful
ρ134-400-210-12-3--30000    complex faithful
ρ144-4001-202-100--3-300    complex faithful
ρ154-4001-202-100-3--300    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

3243
4556
3361
0001
,
6211
2661
0010
0002
,
0321
1154
4353
3341
,
0110
1010
0060
0001
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

Export

Subgroup lattice of C32⋊D8 in TeX
Character table of C32⋊D8 in TeX

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