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## G = SD16.D6order 192 = 26·3

### The non-split extension by SD16 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — SD16.D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — Q8.15D6 — SD16.D6
 Lower central C3 — C6 — C12 — SD16.D6
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for SD16.D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=a3, cac-1=dad-1=a5, cbc-1=a4b, bd=db, dcd-1=a4c-1 >

Subgroups: 608 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8○D4, C2×Q16, C4○D8, C8.C22, C8.C22, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, D42S3, S3×Q8, S3×Q8, Q83S3, Q83S3, C6×Q8, C3×C4○D4, Q8○D8, D12.C4, C8.D6, D4.D6, Q8.7D6, S3×Q16, Q16⋊S3, C2×C3⋊Q16, Q8.13D6, C3×C8.C22, Q8.15D6, Q8○D12, SD16.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, Q8○D8, C2×S3×D4, SD16.D6

Smallest permutation representation of SD16.D6
On 96 points
Generators in S96
(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 61)(2 64)(3 59)(4 62)(5 57)(6 60)(7 63)(8 58)(9 79)(10 74)(11 77)(12 80)(13 75)(14 78)(15 73)(16 76)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(25 47)(26 42)(27 45)(28 48)(29 43)(30 46)(31 41)(32 44)(49 71)(50 66)(51 69)(52 72)(53 67)(54 70)(55 65)(56 68)(81 91)(82 94)(83 89)(84 92)(85 95)(86 90)(87 93)(88 96)
(1 49 82 38 75 42)(2 54 83 35 76 47)(3 51 84 40 77 44)(4 56 85 37 78 41)(5 53 86 34 79 46)(6 50 87 39 80 43)(7 55 88 36 73 48)(8 52 81 33 74 45)(9 26 57 71 90 23)(10 31 58 68 91 20)(11 28 59 65 92 17)(12 25 60 70 93 22)(13 30 61 67 94 19)(14 27 62 72 95 24)(15 32 63 69 96 21)(16 29 64 66 89 18)
(1 25 5 29)(2 30 6 26)(3 27 7 31)(4 32 8 28)(9 50 13 54)(10 55 14 51)(11 52 15 56)(12 49 16 53)(17 85 21 81)(18 82 22 86)(19 87 23 83)(20 84 24 88)(33 96 37 92)(34 93 38 89)(35 90 39 94)(36 95 40 91)(41 59 45 63)(42 64 46 60)(43 61 47 57)(44 58 48 62)(65 78 69 74)(66 75 70 79)(67 80 71 76)(68 77 72 73)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 24A 24B order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 12 12 12 12 12 24 24 size 1 1 2 4 6 6 12 2 2 2 4 4 4 6 6 12 12 12 2 4 8 4 4 6 6 12 4 4 8 8 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 S3×D4 S3×D4 Q8○D8 SD16.D6 kernel SD16.D6 D12.C4 C8.D6 D4.D6 Q8.7D6 S3×Q16 Q16⋊S3 C2×C3⋊Q16 Q8.13D6 C3×C8.C22 Q8.15D6 Q8○D12 C8.C22 Dic6 D12 C3⋊D4 M4(2) SD16 Q16 C2×Q8 C4○D4 C4 C22 C3 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 2 2 1 1 1 1 2 1

Matrix representation of SD16.D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 19 71 0 34 0 0 1 0 56 17 0 0 55 1 72 57 0 0 71 57 72 55
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 41 32 0 0 0 0 57 32 0 0 0 0 32 57 16 16 0 0 16 57 16 57
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 32 0 32 32 0 0 16 0 0 32 0 0 0 57 57 57 0 0 57 16 57 57
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 17 0 2 71 0 0 1 17 2 0 0 0 56 1 55 1 0 0 55 1 72 57

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,19,1,55,71,0,0,71,0,1,57,0,0,0,56,72,72,0,0,34,17,57,55],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,57,32,16,0,0,32,32,57,57,0,0,0,0,16,16,0,0,0,0,16,57],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,32,16,0,57,0,0,0,0,57,16,0,0,32,0,57,57,0,0,32,32,57,57],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,17,1,56,55,0,0,0,17,1,1,0,0,2,2,55,72,0,0,71,0,1,57] >;

SD16.D6 in GAP, Magma, Sage, TeX

{\rm SD}_{16}.D_6
% in TeX

G:=Group("SD16.D6");
// GroupNames label

G:=SmallGroup(192,1338);
// by ID

G=gap.SmallGroup(192,1338);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,570,185,136,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=a^3,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations

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