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## G = SD16⋊C8order 128 = 27

### The semidirect product of SD16 and C8 acting via C8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — SD16⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×D4 — SD16⋊C8
 Lower central C1 — C2 — C4 — SD16⋊C8
 Upper central C1 — C2×C4 — C4×C8 — SD16⋊C8
 Jennings C1 — C22 — C22 — C42 — SD16⋊C8

Generators and relations for SD16⋊C8
G = < a,b,c | a8=b2=c8=1, bab=a3, cac-1=a5, bc=cb >

Subgroups: 152 in 88 conjugacy classes, 50 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×5], C22, C22 [×4], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×5], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C8×D4, C4×SD16, C8×Q8, SD16⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8⋊C22, C8.C22, C8×D4, SD16⋊C4, C8.26D4, SD16⋊C8

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 8A ··· 8H 8I ··· 8X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 D4 C4○D4 C8○D4 C8⋊C22 C8.C22 C8.26D4 kernel SD16⋊C8 C8⋊C8 D4⋊C8 Q8⋊C8 C8⋊1C8 C8×D4 C4×SD16 C8×Q8 D4⋊C4 Q8⋊C4 C4.Q8 C2×SD16 SD16 C2×C8 C2×C4 C4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 16 2 2 4 1 1 2

Matrix representation of SD16⋊C8 in GL6(𝔽17)

 1 4 0 0 0 0 8 16 0 0 0 0 0 0 0 4 0 16 0 0 15 13 9 1 0 0 0 1 0 13 0 0 8 16 2 4
,
 16 0 0 0 0 0 9 1 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(17))| [1,8,0,0,0,0,4,16,0,0,0,0,0,0,0,15,0,8,0,0,4,13,1,16,0,0,0,9,0,2,0,0,16,1,13,4],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD16⋊C8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,310);
// by ID

G=gap.SmallGroup(128,310);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,100,1123,570,136,172]);
// Polycyclic

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

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