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## G = C6.SD16order 96 = 25·3

### 2nd non-split extension by C6 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6.SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C6.SD16
 Lower central C3 — C6 — C12 — C6.SD16
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C6.SD16
G = < a,b,c | a6=b8=1, c2=a3b4, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Character table of C6.SD16

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 12 12 2 2 2 6 6 6 6 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 i -i 1 -1 -1 -1 1 -i i -i i -1 i i 1 -i -i linear of order 4 ρ6 1 1 -1 -1 1 -1 1 i -i -1 1 -1 -1 1 i -i i -i -1 i i 1 -i -i linear of order 4 ρ7 1 1 -1 -1 1 -1 1 -i i -1 1 -1 -1 1 -i i -i i -1 -i -i 1 i i linear of order 4 ρ8 1 1 -1 -1 1 -1 1 -i i 1 -1 -1 -1 1 i -i i -i -1 -i -i 1 i i linear of order 4 ρ9 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 2 -2 -2 0 0 0 0 2 2 2 0 0 0 0 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 2 2 -2 -2 0 0 -1 -1 -1 0 0 0 0 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 -1 2 -2 0 0 0 0 1 1 -1 0 0 0 0 -1 √3 -√3 1 -√3 √3 orthogonal lifted from D12 ρ14 2 2 -2 -2 -1 2 -2 0 0 0 0 1 1 -1 0 0 0 0 -1 -√3 √3 1 √3 -√3 orthogonal lifted from D12 ρ15 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ17 2 2 -2 -2 -1 -2 2 -2i 2i 0 0 1 1 -1 0 0 0 0 1 i i -1 -i -i complex lifted from C4×S3 ρ18 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 √-3 -√-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 2 -2 -2 -1 -2 2 2i -2i 0 0 1 1 -1 0 0 0 0 1 -i -i -1 i i complex lifted from C4×S3 ρ21 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 -√-3 √-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ23 4 -4 -4 4 -2 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ24 4 -4 4 -4 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of C6.SD16
Regular action on 96 points
Generators in S96
(1 75 93 23 27 61)(2 62 28 24 94 76)(3 77 95 17 29 63)(4 64 30 18 96 78)(5 79 89 19 31 57)(6 58 32 20 90 80)(7 73 91 21 25 59)(8 60 26 22 92 74)(9 48 34 55 67 83)(10 84 68 56 35 41)(11 42 36 49 69 85)(12 86 70 50 37 43)(13 44 38 51 71 87)(14 88 72 52 39 45)(15 46 40 53 65 81)(16 82 66 54 33 47)
(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)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 55 19 13)(2 16 20 50)(3 53 21 11)(4 14 22 56)(5 51 23 9)(6 12 24 54)(7 49 17 15)(8 10 18 52)(25 42 77 65)(26 68 78 45)(27 48 79 71)(28 66 80 43)(29 46 73 69)(30 72 74 41)(31 44 75 67)(32 70 76 47)(33 58 86 94)(34 89 87 61)(35 64 88 92)(36 95 81 59)(37 62 82 90)(38 93 83 57)(39 60 84 96)(40 91 85 63)

G:=sub<Sym(96)| (1,75,93,23,27,61)(2,62,28,24,94,76)(3,77,95,17,29,63)(4,64,30,18,96,78)(5,79,89,19,31,57)(6,58,32,20,90,80)(7,73,91,21,25,59)(8,60,26,22,92,74)(9,48,34,55,67,83)(10,84,68,56,35,41)(11,42,36,49,69,85)(12,86,70,50,37,43)(13,44,38,51,71,87)(14,88,72,52,39,45)(15,46,40,53,65,81)(16,82,66,54,33,47), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,42,77,65)(26,68,78,45)(27,48,79,71)(28,66,80,43)(29,46,73,69)(30,72,74,41)(31,44,75,67)(32,70,76,47)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63)>;

G:=Group( (1,75,93,23,27,61)(2,62,28,24,94,76)(3,77,95,17,29,63)(4,64,30,18,96,78)(5,79,89,19,31,57)(6,58,32,20,90,80)(7,73,91,21,25,59)(8,60,26,22,92,74)(9,48,34,55,67,83)(10,84,68,56,35,41)(11,42,36,49,69,85)(12,86,70,50,37,43)(13,44,38,51,71,87)(14,88,72,52,39,45)(15,46,40,53,65,81)(16,82,66,54,33,47), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,42,77,65)(26,68,78,45)(27,48,79,71)(28,66,80,43)(29,46,73,69)(30,72,74,41)(31,44,75,67)(32,70,76,47)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63) );

G=PermutationGroup([(1,75,93,23,27,61),(2,62,28,24,94,76),(3,77,95,17,29,63),(4,64,30,18,96,78),(5,79,89,19,31,57),(6,58,32,20,90,80),(7,73,91,21,25,59),(8,60,26,22,92,74),(9,48,34,55,67,83),(10,84,68,56,35,41),(11,42,36,49,69,85),(12,86,70,50,37,43),(13,44,38,51,71,87),(14,88,72,52,39,45),(15,46,40,53,65,81),(16,82,66,54,33,47)], [(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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,55,19,13),(2,16,20,50),(3,53,21,11),(4,14,22,56),(5,51,23,9),(6,12,24,54),(7,49,17,15),(8,10,18,52),(25,42,77,65),(26,68,78,45),(27,48,79,71),(28,66,80,43),(29,46,73,69),(30,72,74,41),(31,44,75,67),(32,70,76,47),(33,58,86,94),(34,89,87,61),(35,64,88,92),(36,95,81,59),(37,62,82,90),(38,93,83,57),(39,60,84,96),(40,91,85,63)])

Matrix representation of C6.SD16 in GL5(𝔽73)

 72 0 0 0 0 0 72 72 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 27 0 0 0 0 0 54 68 0 0 0 14 19 0 0 0 0 0 0 4 0 0 0 55 12
,
 27 0 0 0 0 0 30 60 0 0 0 13 43 0 0 0 0 0 11 66 0 0 0 38 62

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,54,14,0,0,0,68,19,0,0,0,0,0,0,55,0,0,0,4,12],[27,0,0,0,0,0,30,13,0,0,0,60,43,0,0,0,0,0,11,38,0,0,0,66,62] >;

C6.SD16 in GAP, Magma, Sage, TeX

C_6.{\rm SD}_{16}
% in TeX

G:=Group("C6.SD16");
// GroupNames label

G:=SmallGroup(96,17);
// by ID

G=gap.SmallGroup(96,17);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=1,c^2=a^3*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

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