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## G = SD16⋊3D4order 128 = 27

### 3rd semidirect product of SD16 and D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — SD16⋊3D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C2×C4○D4 — C2×C8.C22 — SD16⋊3D4
 Lower central C1 — C2 — C2×C4 — SD16⋊3D4
 Upper central C1 — C22 — C4×D4 — SD16⋊3D4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊3D4

Generators and relations for SD163D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 448 in 236 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×10], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×2], D4 [×7], Q8 [×2], Q8 [×13], C23 [×2], C23, C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×4], SD16 [×4], Q16 [×10], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×4], C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×M4(2) [×2], C2×SD16, C2×SD16 [×2], C2×Q16 [×6], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4 [×2], C86D4, C4×SD16, C22⋊Q16 [×2], D4.7D4 [×2], D4.D4, C42Q16, C8.D4 [×2], C4⋊Q16, D46D4, D4×Q8, C2×C8.C22 [×2], SD163D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8.C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8.C22, Q8○D8, SD163D4

Character table of SD163D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 4 2 2 2 2 4 4 4 4 4 8 8 8 8 8 8 4 4 4 4 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 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 2 2 2 0 0 2 -2 2 -2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 0 2 -2 0 -2 0 2 0 2 0 0 0 -2 0 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 -2 -2 2 -2 2 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 0 -2 2 0 -2 0 2 0 2 0 0 0 -2 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 -2 0 0 2 -2 -2 -2 -2 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 -2 2 -2 0 -2 2 0 -2 0 2 0 -2 0 0 0 2 0 0 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ23 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 -2 2 -2 0 2 -2 0 -2 0 2 0 -2 0 0 0 2 0 0 0 0 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ25 4 -4 4 -4 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ26 4 -4 -4 4 0 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ27 4 -4 -4 4 0 0 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of SD163D4
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 10)(2 13)(3 16)(4 11)(5 14)(6 9)(7 12)(8 15)(17 43)(18 46)(19 41)(20 44)(21 47)(22 42)(23 45)(24 48)(25 35)(26 38)(27 33)(28 36)(29 39)(30 34)(31 37)(32 40)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)
(1 24 10 48)(2 17 11 41)(3 18 12 42)(4 19 13 43)(5 20 14 44)(6 21 15 45)(7 22 16 46)(8 23 9 47)(25 52 39 63)(26 53 40 64)(27 54 33 57)(28 55 34 58)(29 56 35 59)(30 49 36 60)(31 50 37 61)(32 51 38 62)
(1 56)(2 53)(3 50)(4 55)(5 52)(6 49)(7 54)(8 51)(9 62)(10 59)(11 64)(12 61)(13 58)(14 63)(15 60)(16 57)(17 26)(18 31)(19 28)(20 25)(21 30)(22 27)(23 32)(24 29)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)

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

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

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

Matrix representation of SD163D4 in GL8(𝔽17)

 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 13 13 14 14 0 0 0 0 4 13 3 14 0 0 0 0 14 14 4 4 0 0 0 0 3 14 13 4
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16
,
 13 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,4,14,3,0,0,0,0,13,13,14,14,0,0,0,0,14,3,4,13,0,0,0,0,14,14,4,4],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0] >;

SD163D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3D_4
% in TeX

G:=Group("SD16:3D4");
// GroupNames label

G:=SmallGroup(128,2008);
// by ID

G=gap.SmallGroup(128,2008);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,352,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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