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## G = C32⋊SD32order 288 = 25·32

### The semidirect product of C32 and SD32 acting via SD32/C4=D4

Aliases: C32⋊SD32, C4.2S3≀C2, (C3×C6).2D8, (C3×C12).6D4, C322D8.C2, C2.4(C32⋊D8), C322C162C2, C322Q161C2, C324C8.2C22, SmallGroup(288,383)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊4C8 — C32⋊SD32
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2D8 — C32⋊SD32
 Lower central C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊SD32
 Upper central C1 — C2 — C4

Generators and relations for C32⋊SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=b-1, dad=a-1, cbc-1=a, bd=db, dcd=c7 >

Character table of C32⋊SD32

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 16A 16B 16C 16D size 1 1 24 4 4 2 24 4 4 24 24 18 18 8 8 24 24 18 18 18 18 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 2 2 2 0 2 2 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 2 2 -2 0 2 2 0 0 0 0 -2 -2 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ7 2 2 0 2 2 -2 0 2 2 0 0 0 0 -2 -2 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ8 2 -2 0 2 2 0 0 -2 -2 0 0 √2 -√2 0 0 0 0 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ9 2 -2 0 2 2 0 0 -2 -2 0 0 -√2 √2 0 0 0 0 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32 ρ10 2 -2 0 2 2 0 0 -2 -2 0 0 √2 -√2 0 0 0 0 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ11 2 -2 0 2 2 0 0 -2 -2 0 0 -√2 √2 0 0 0 0 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ12 4 4 -2 -2 1 4 0 -2 1 1 1 0 0 -2 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 1 -2 4 2 1 -2 0 0 0 0 1 -2 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 2 -2 1 4 0 -2 1 -1 -1 0 0 -2 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 0 1 -2 4 -2 1 -2 0 0 0 0 1 -2 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 0 1 -2 -4 0 1 -2 0 0 0 0 -1 2 √-3 -√-3 0 0 0 0 complex lifted from C32⋊D8 ρ17 4 4 0 1 -2 -4 0 1 -2 0 0 0 0 -1 2 -√-3 √-3 0 0 0 0 complex lifted from C32⋊D8 ρ18 4 4 0 -2 1 -4 0 -2 1 -√-3 √-3 0 0 2 -1 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ19 4 4 0 -2 1 -4 0 -2 1 √-3 -√-3 0 0 2 -1 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ20 8 -8 0 -4 2 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ21 8 -8 0 2 -4 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊SD32
On 48 points
Generators in S48
(1 39 32)(3 18 41)(5 43 20)(7 22 45)(9 47 24)(11 26 33)(13 35 28)(15 30 37)
(2 40 17)(4 19 42)(6 44 21)(8 23 46)(10 48 25)(12 27 34)(14 36 29)(16 31 38)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 46)(18 37)(19 44)(20 35)(21 42)(22 33)(23 40)(24 47)(25 38)(26 45)(27 36)(28 43)(29 34)(30 41)(31 48)(32 39)

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

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

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

Matrix representation of C32⋊SD32 in GL6(𝔽97)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0
,
 34 77 0 0 0 0 10 54 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 41 15 0 0 0 0 82 56 0 0
,
 96 95 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 41 15 0 0 0 0 82 56

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[34,10,0,0,0,0,77,54,0,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56,0,0,0,1,0,0,0,0,1,0,0,0],[96,0,0,0,0,0,95,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;

C32⋊SD32 in GAP, Magma, Sage, TeX

C_3^2\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C3^2:SD32");
// GroupNames label

G:=SmallGroup(288,383);
// by ID

G=gap.SmallGroup(288,383);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,120,254,135,142,675,346,80,2693,2028,691,797,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=b^-1,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^7>;
// generators/relations

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