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## G = C8⋊5SD16order 128 = 27

### 5th 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⋊5SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8⋊5SD16
 Lower central C1 — C22 — C42 — C8⋊5SD16
 Upper central C1 — C22 — C42 — C8⋊5SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊5SD16

Generators and relations for C85SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a5, cac=a3, cbc=b3 >

Subgroups: 256 in 99 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C41D4, C4⋊Q8, C2×SD16, C8⋊C8, C4.4D8, C4.SD16, C85D4, C83Q8, C82Q8, C85SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C41D4, C2×SD16, C8⋊C22, C8.C22, C85D4, C83D4, C8.2D4, C85SD16

Character table of C85SD16

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

Smallest permutation representation of C85SD16
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 63 17 46 11 25 53 35)(2 60 18 43 12 30 54 40)(3 57 19 48 13 27 55 37)(4 62 20 45 14 32 56 34)(5 59 21 42 15 29 49 39)(6 64 22 47 16 26 50 36)(7 61 23 44 9 31 51 33)(8 58 24 41 10 28 52 38)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 35)(26 38)(27 33)(28 36)(29 39)(30 34)(31 37)(32 40)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)

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

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

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

Matrix representation of C85SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 13 5 12 0 0 4 4 5 5 0 0 12 5 13 4 0 0 12 12 13 13
,
 5 12 0 0 0 0 5 5 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
,
 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 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,12,12,0,0,13,4,5,12,0,0,5,5,13,13,0,0,12,5,4,13],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C85SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C8:5SD16");
// GroupNames label

G:=SmallGroup(128,446);
// by ID

G=gap.SmallGroup(128,446);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,64,422,387,100,1123,136,2804,172]);
// Polycyclic

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

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