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## G = C32⋊2D8order 144 = 24·32

### 1st semidirect product of C32 and D8 acting via D8/C4=C22

Aliases: C322D8, D122S3, C12.9D6, C4.8S32, (C3×C6).6D4, C32(D4⋊S3), (C3×D12)⋊1C2, C324C81C2, C6.7(C3⋊D4), (C3×C12).1C22, C2.3(D6⋊S3), SmallGroup(144,56)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊2D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8
 Lower central C32 — C3×C6 — C3×C12 — C32⋊2D8
 Upper central C1 — C2 — C4

Generators and relations for C322D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Character table of C322D8

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D size 1 1 12 12 2 2 4 2 2 2 4 12 12 12 12 18 18 4 4 4 4 ρ1 1 1 1 1 1 1 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 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 -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 -1 -1 1 1 1 1 linear of order 2 ρ5 2 2 0 0 2 2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 -2 0 2 -1 -1 2 -1 2 -1 0 0 1 1 0 0 2 -1 -1 -1 orthogonal lifted from D6 ρ7 2 2 2 0 2 -1 -1 2 -1 2 -1 0 0 -1 -1 0 0 2 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 -1 2 -1 2 2 -1 -1 -1 -1 0 0 0 0 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 2 0 -2 -1 2 -1 2 2 -1 -1 1 1 0 0 0 0 -1 2 -1 -1 orthogonal lifted from D6 ρ10 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ11 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 0 2 -1 -1 -2 -1 2 -1 0 0 -√-3 √-3 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ13 2 2 0 0 2 -1 -1 -2 -1 2 -1 0 0 √-3 -√-3 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ14 2 2 0 0 -1 2 -1 -2 2 -1 -1 √-3 -√-3 0 0 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 0 -1 2 -1 -2 2 -1 -1 -√-3 √-3 0 0 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ16 4 -4 0 0 4 -2 -2 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 0 0 -2 4 -2 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ18 4 4 0 0 -2 -2 1 4 -2 -2 1 0 0 0 0 0 0 -2 -2 1 1 orthogonal lifted from S32 ρ19 4 4 0 0 -2 -2 1 -4 -2 -2 1 0 0 0 0 0 0 2 2 -1 -1 symplectic lifted from D6⋊S3, Schur index 2 ρ20 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 0 0 0 0 3i -3i complex faithful ρ21 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 0 0 0 0 -3i 3i complex faithful

Smallest permutation representation of C322D8
On 48 points
Generators in S48
(1 35 46)(2 47 36)(3 37 48)(4 41 38)(5 39 42)(6 43 40)(7 33 44)(8 45 34)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(1 46 35)(2 36 47)(3 48 37)(4 38 41)(5 42 39)(6 40 43)(7 44 33)(8 34 45)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

G:=sub<Sym(48)| (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)>;

G:=Group( (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43) );

G=PermutationGroup([(1,35,46),(2,47,36),(3,37,48),(4,41,38),(5,39,42),(6,43,40),(7,33,44),(8,45,34),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(1,46,35),(2,36,47),(3,48,37),(4,38,41),(5,42,39),(6,40,43),(7,44,33),(8,34,45),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)])

C322D8 is a maximal subgroup of
C32⋊D16  C32⋊SD32  C244D6  C246D6  D12.2D6  D12.30D6  D1220D6  S3×D4⋊S3  D129D6  D126D6  D12.12D6  D36⋊S3  He33D8  C336D8  C339D8
C322D8 is a maximal quotient of
C322D16  D24.S3  C322Q32  D123Dic3  C12.8Dic6  D36⋊S3  He32D8  C336D8  C339D8

Matrix representation of C322D8 in GL4(𝔽5) generated by

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

C322D8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2D_8
% in TeX

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

G:=SmallGroup(144,56);
// by ID

G=gap.SmallGroup(144,56);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,490,3461]);
// Polycyclic

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

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