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## G = C8⋊SD16order 128 = 27

### 1st semidirect product of C8 and SD16 acting via SD16/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8⋊4Q8 — C8⋊SD16
 Lower central C1 — C22 — C42 — C8⋊SD16
 Upper central C1 — C22 — C42 — C8⋊SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊SD16

Generators and relations for C8⋊SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b3 >

Subgroups: 216 in 84 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×Q8, C41D4, C4⋊Q8, C2×SD16, C4.D8, C4.10D8, C81C8, C84Q8, C4⋊SD16, Q8⋊Q8, C85D4, C8⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C8.C22, C4⋊SD16, C8⋊D4, D4.3D4, C8⋊SD16

Character table of C8⋊SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 16 2 2 2 2 4 8 8 16 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 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 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 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 1 1 linear of order 2 ρ5 1 1 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 ρ6 1 1 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 ρ7 1 1 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 ρ8 1 1 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 ρ9 2 2 2 2 0 2 -2 2 -2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 -2 2 -2 -2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from C4○D4 ρ14 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from C4○D4 ρ15 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 -2 2 0 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ16 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 2 -2 0 √-2 √-2 -√-2 -√-2 0 0 complex lifted from SD16 ρ17 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 2 -2 0 -√-2 -√-2 √-2 √-2 0 0 complex lifted from SD16 ρ18 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 -2 2 0 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ19 4 -4 4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 -4 4 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ21 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ22 4 4 -4 -4 0 0 0 0 0 0 0 0 0 2√-2 0 0 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 -2√-2 0 0 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4

Smallest permutation representation of C8⋊SD16
On 64 points
Generators in S64
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 29 35 59 55 43 21 12)(2 28 36 58 56 42 22 11)(3 27 37 57 49 41 23 10)(4 26 38 64 50 48 24 9)(5 25 39 63 51 47 17 16)(6 32 40 62 52 46 18 15)(7 31 33 61 53 45 19 14)(8 30 34 60 54 44 20 13)
(2 4)(3 7)(6 8)(9 42)(10 45)(11 48)(12 43)(13 46)(14 41)(15 44)(16 47)(17 39)(18 34)(19 37)(20 40)(21 35)(22 38)(23 33)(24 36)(25 63)(26 58)(27 61)(28 64)(29 59)(30 62)(31 57)(32 60)(49 53)(50 56)(52 54)

G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,29,35,59,55,43,21,12)(2,28,36,58,56,42,22,11)(3,27,37,57,49,41,23,10)(4,26,38,64,50,48,24,9)(5,25,39,63,51,47,17,16)(6,32,40,62,52,46,18,15)(7,31,33,61,53,45,19,14)(8,30,34,60,54,44,20,13), (2,4)(3,7)(6,8)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(49,53)(50,56)(52,54)>;

G:=Group( (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,29,35,59,55,43,21,12)(2,28,36,58,56,42,22,11)(3,27,37,57,49,41,23,10)(4,26,38,64,50,48,24,9)(5,25,39,63,51,47,17,16)(6,32,40,62,52,46,18,15)(7,31,33,61,53,45,19,14)(8,30,34,60,54,44,20,13), (2,4)(3,7)(6,8)(9,42)(10,45)(11,48)(12,43)(13,46)(14,41)(15,44)(16,47)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(49,53)(50,56)(52,54) );

G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,29,35,59,55,43,21,12),(2,28,36,58,56,42,22,11),(3,27,37,57,49,41,23,10),(4,26,38,64,50,48,24,9),(5,25,39,63,51,47,17,16),(6,32,40,62,52,46,18,15),(7,31,33,61,53,45,19,14),(8,30,34,60,54,44,20,13)], [(2,4),(3,7),(6,8),(9,42),(10,45),(11,48),(12,43),(13,46),(14,41),(15,44),(16,47),(17,39),(18,34),(19,37),(20,40),(21,35),(22,38),(23,33),(24,36),(25,63),(26,58),(27,61),(28,64),(29,59),(30,62),(31,57),(32,60),(49,53),(50,56),(52,54)]])

Matrix representation of C8⋊SD16 in GL8(𝔽17)

 8 11 16 16 0 0 0 0 5 8 0 16 0 0 0 0 6 5 14 11 0 0 0 0 12 1 12 4 0 0 0 0 0 0 0 0 0 13 6 6 0 0 0 0 0 0 11 0 0 0 0 0 11 14 0 0 0 0 0 0 3 0 13 0
,
 0 0 1 0 0 0 0 0 1 1 1 2 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 7 15 15 0 0 0 0 1 0 0 0 0 0 0 0 16 7 10 10
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 16 16 15 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 15 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 10 0 16 16

G:=sub<GL(8,GF(17))| [8,5,6,12,0,0,0,0,11,8,5,1,0,0,0,0,16,0,14,12,0,0,0,0,16,16,11,4,0,0,0,0,0,0,0,0,0,0,11,3,0,0,0,0,13,0,14,0,0,0,0,0,6,11,0,13,0,0,0,0,6,0,0,0],[0,1,0,0,0,0,0,0,0,1,16,0,0,0,0,0,1,1,0,16,0,0,0,0,0,2,0,16,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,7,0,7,0,0,0,0,1,15,0,10,0,0,0,0,0,15,0,10],[1,0,16,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,15,0,10,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16] >;

C8⋊SD16 in GAP, Magma, Sage, TeX

C_8\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C8:SD16");
// GroupNames label

G:=SmallGroup(128,418);
// by ID

G=gap.SmallGroup(128,418);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,352,1123,136,2804,718,172]);
// Polycyclic

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

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