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G = C8⋊6SD16order 128 = 27

6th 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⋊6SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊C8 — C8⋊6SD16
 Lower central C1 — C22 — C42 — C8⋊6SD16
 Upper central C1 — C22 — C42 — C8⋊6SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊6SD16

Generators and relations for C86SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a5, cac=a-1, cbc=b3 >

Subgroups: 320 in 110 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, C4×C8, C4×C8, D4⋊C4, C4.Q8, C41D4, C4⋊Q8, C2×D8, C2×SD16, C8⋊C8, C4.4D8, C85D4, C84D4, C83Q8, C86SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C41D4, C2×SD16, C8⋊C22, C85D4, C83D4, C86SD16

Character table of C86SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 1 1 16 16 2 2 2 2 2 2 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 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 2 0 0 -2 0 0 -2 2 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 -2 -2 0 0 2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 2 -2 2 -2 -2 -2 0 0 -2 2 0 0 0 2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -2 -2 -2 2 -2 2 0 0 0 0 2 2 0 0 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 -2 0 0 2 0 0 2 -2 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 2 -2 2 -2 -2 -2 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 0 2 0 0 -2 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 0 -2 0 0 2 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 0 -2 0 0 2 0 0 0 √-2 √-2 -√-2 √-2 0 -√-2 √-2 2 -√-2 -√-2 -2 0 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 -√-2 √-2 √-2 -√-2 -2 -√-2 √-2 0 √-2 -√-2 0 2 complex lifted from SD16 ρ19 2 -2 2 -2 0 0 0 -2 0 0 2 0 0 0 √-2 √-2 √-2 -√-2 0 -√-2 -√-2 -2 -√-2 √-2 2 0 complex lifted from SD16 ρ20 2 -2 2 -2 0 0 0 2 0 0 -2 0 0 0 √-2 -√-2 √-2 -√-2 2 √-2 √-2 0 -√-2 -√-2 0 -2 complex lifted from SD16 ρ21 2 -2 2 -2 0 0 0 2 0 0 -2 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 0 -2 0 0 2 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 0 -4 0 4 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 -4 0 4 0 0 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 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 4 -4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of C86SD16
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 59 20 48 35 29 52 15)(2 64 21 45 36 26 53 12)(3 61 22 42 37 31 54 9)(4 58 23 47 38 28 55 14)(5 63 24 44 39 25 56 11)(6 60 17 41 40 30 49 16)(7 57 18 46 33 27 50 13)(8 62 19 43 34 32 51 10)
(2 8)(3 7)(4 6)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 56)(33 37)(34 36)(38 40)(41 58)(42 57)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)

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

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

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

Matrix representation of C86SD16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 10 7 0 0 5 0 0 7 0 0 0 12 5 12 0 0 12 5 5 12
,
 5 12 0 0 0 0 5 5 0 0 0 0 0 0 1 0 15 0 0 0 0 0 16 1 0 0 0 0 16 0 0 0 0 1 16 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 1 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,12,0,0,0,0,12,5,0,0,10,0,5,5,0,0,7,7,12,12],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C86SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("C8:6SD16");
// GroupNames label

G:=SmallGroup(128,447);
// by ID

G=gap.SmallGroup(128,447);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,100,1123,136,2804,172]);
// Polycyclic

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

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