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## G = C8⋊13SD16order 128 = 27

### 1st semidirect product of C8 and SD16 acting via SD16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊13SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8×Q8 — C8⋊13SD16
 Lower central C1 — C22 — C42 — C8⋊13SD16
 Upper central C1 — C22 — C42 — C8⋊13SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊13SD16

Generators and relations for C813SD16
G = < a,b,c | a8=b8=c2=1, bab-1=cac=a-1, cbc=b3 >

Subgroups: 272 in 94 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×6], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×2], D4 [×10], Q8 [×2], Q8, C23 [×2], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×3], D8 [×4], SD16 [×4], C2×D4 [×6], C2×Q8, C4×C8, C4×C8, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C4×Q8, C41D4 [×2], C2×D8 [×2], C2×SD16 [×2], C4.D8 [×2], C81C8, C8×Q8, C4⋊SD16 [×2], C84D4, C813SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C2×SD16, C4○D8, C8⋊C22, C4⋊SD16, C87D4, D4.4D4, C813SD16

Character table of C813SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N size 1 1 1 1 16 16 2 2 2 2 4 4 4 4 4 2 2 2 2 4 4 4 4 4 4 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 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 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 -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 -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 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 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 -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 -1 -1 1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 0 0 -2 -2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 -2 -2 0 0 -2 0 0 2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 -2 2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 2 2 -2 -2 0 0 -2 0 0 -2 -2 -2 -2 0 0 0 2 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 0 0 0 -2 2 0 2 0 0 -2 -√2 √2 √2 -√2 √2 -√2 -√2 √2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 -2 -2 0 0 0 0 -2 2 0 -2 0 0 2 -√2 √2 √2 -√2 -√2 √2 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ15 2 2 -2 -2 0 0 0 0 -2 2 0 2 0 0 -2 √2 -√2 -√2 √2 -√2 √2 √2 -√2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 0 0 0 -2 2 0 -2 0 0 2 √2 -√2 -√2 √2 √2 -√2 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 2 0 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 2i 2i -2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 0 0 0 0 2 -2 2i 0 0 -2i 0 -√2 √2 √2 -√2 √-2 -√-2 √-2 -√2 √2 -√-2 0 0 0 0 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 0 0 0 2 -2 -2i 0 0 2i 0 √2 -√2 -√2 √2 √-2 -√-2 √-2 √2 -√2 -√-2 0 0 0 0 complex lifted from C4○D8 ρ20 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ21 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ22 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ23 2 2 -2 -2 0 0 0 0 2 -2 2i 0 0 -2i 0 √2 -√2 -√2 √2 -√-2 √-2 -√-2 √2 -√2 √-2 0 0 0 0 complex lifted from C4○D8 ρ24 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ25 2 2 -2 -2 0 0 0 0 2 -2 -2i 0 0 2i 0 -√2 √2 √2 -√2 -√-2 √-2 -√-2 -√2 √2 √-2 0 0 0 0 complex lifted from C4○D8 ρ26 2 2 2 2 0 0 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 -2i -2i 2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ27 4 -4 4 -4 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ29 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4.4D4

Smallest permutation representation of C813SD16
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 44 19 10 57 29 38 55)(2 43 20 9 58 28 39 54)(3 42 21 16 59 27 40 53)(4 41 22 15 60 26 33 52)(5 48 23 14 61 25 34 51)(6 47 24 13 62 32 35 50)(7 46 17 12 63 31 36 49)(8 45 18 11 64 30 37 56)
(2 8)(3 7)(4 6)(9 45)(10 44)(11 43)(12 42)(13 41)(14 48)(15 47)(16 46)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 51)(26 50)(27 49)(28 56)(29 55)(30 54)(31 53)(32 52)(58 64)(59 63)(60 62)

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

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

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

Matrix representation of C813SD16 in GL4(𝔽17) generated by

 0 6 0 0 14 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 1 16 0 0 0 0 7 10 0 0 12 0
,
 1 0 0 0 1 16 0 0 0 0 1 0 0 0 1 16
G:=sub<GL(4,GF(17))| [0,14,0,0,6,6,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,16,0,0,0,0,7,12,0,0,10,0],[1,1,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C813SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{13}{\rm SD}_{16}
% in TeX

G:=Group("C8:13SD16");
// GroupNames label

G:=SmallGroup(128,400);
// by ID

G=gap.SmallGroup(128,400);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,352,1123,136,2804,718,172]);
// Polycyclic

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

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