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## G = SD16⋊8D4order 128 = 27

### 4th semidirect product of SD16 and D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — SD16⋊8D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C2×C4○D4 — C2×C4○D8 — SD16⋊8D4
 Lower central C1 — C2 — C2×C4 — SD16⋊8D4
 Upper central C1 — C22 — C4×D4 — SD16⋊8D4
 Jennings C1 — C2 — C2 — C2×C4 — SD16⋊8D4

Generators and relations for SD168D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 472 in 239 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×13], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×20], D4 [×2], D4 [×12], Q8 [×2], Q8 [×9], C23 [×2], C23 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×4], SD16 [×5], Q16 [×7], C22×C4 [×2], C22×C4 [×5], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×9], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16 [×4], C2×Q16 [×4], C4○D8 [×4], C8.C22 [×4], C22×Q8, C2×C4○D4 [×3], C89D4, SD16⋊C4, D4⋊D4, C22⋊Q16, D4.7D4 [×2], C42Q16, D4.2D4, C8.18D4, C8⋊D4, C8.2D4, D46D4, Q85D4, C2×C4○D8, C2×C8.C22, SD168D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8⋊C22, Q8○D8, SD168D4

Character table of SD168D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 4 8 2 2 2 2 4 4 4 4 4 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 0 -2 2 0 0 0 -2 2 0 -2 0 0 0 2 0 0 0 0 0 -2 0 2 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 0 2 -2 0 0 0 -2 2 0 -2 0 0 0 2 0 0 0 0 0 2 0 -2 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 2 0 -2 -2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 0 0 -2 0 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 0 2 -2 0 0 0 -2 2 0 2 0 0 0 -2 0 0 0 0 0 -2 0 2 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 2 0 0 2 0 -2 -2 -2 -2 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 2 2 0 0 -2 0 2 -2 -2 2 0 -2 2 -2 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 0 0 -2 2 0 2 0 0 0 -2 0 0 0 0 0 2 0 -2 0 0 0 orthogonal lifted from D4 ρ25 4 -4 4 -4 0 0 0 0 0 0 4 -4 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ27 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 0 -2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ28 4 -4 -4 4 0 0 0 0 0 -4i 0 0 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22 ρ29 4 -4 -4 4 0 0 0 0 0 4i 0 0 -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊C22

Smallest permutation representation of SD168D4
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 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 64)(26 59)(27 62)(28 57)(29 60)(30 63)(31 58)(32 61)(33 56)(34 51)(35 54)(36 49)(37 52)(38 55)(39 50)(40 53)
(1 27 45 53)(2 32 46 50)(3 29 47 55)(4 26 48 52)(5 31 41 49)(6 28 42 54)(7 25 43 51)(8 30 44 56)(9 61 22 39)(10 58 23 36)(11 63 24 33)(12 60 17 38)(13 57 18 35)(14 62 19 40)(15 59 20 37)(16 64 21 34)
(1 37)(2 34)(3 39)(4 36)(5 33)(6 38)(7 35)(8 40)(9 55)(10 52)(11 49)(12 54)(13 51)(14 56)(15 53)(16 50)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 63)(42 60)(43 57)(44 62)(45 59)(46 64)(47 61)(48 58)

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

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

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

Matrix representation of SD168D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 14 3 0 0 0 0 14 14 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 14 14 0 0 0 0 14 3 0 0 0 0 0 0 14 14 0 0 0 0 14 3
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 10 0 1 0 0 0 0 10 0 1 0 0 1 0 7 0 0 0 0 1 0 7
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 7 0 0 16 0 10 0 0 0 0 7 0 16 0 0 10 0 1 0

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

SD168D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_8D_4
% in TeX

G:=Group("SD16:8D4");
// GroupNames label

G:=SmallGroup(128,2001);
// by ID

G=gap.SmallGroup(128,2001);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,c*a*c^-1=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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