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## G = D6⋊2Q16order 192 = 26·3

### 2nd semidirect product of D6 and Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D6⋊2Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — S3×C2×C8 — D6⋊2Q16
 Lower central C3 — C6 — C2×C12 — D6⋊2Q16
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for D62Q16
G = < a,b,c,d | a6=b2=c8=1, d2=c4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 320 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, S3×C8, Dic12, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C8.18D4, C6.SD16, C3×C2.D8, C4.D12, S3×C2×C8, C2×Dic12, D62Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×Q16, C4○D8, C2×D12, S3×D4, Q83S3, C8.18D4, C12⋊D4, D83S3, S3×Q16, D62Q16

Smallest permutation representation of D62Q16
On 96 points
Generators in S96
(1 31 17 45 82 95)(2 32 18 46 83 96)(3 25 19 47 84 89)(4 26 20 48 85 90)(5 27 21 41 86 91)(6 28 22 42 87 92)(7 29 23 43 88 93)(8 30 24 44 81 94)(9 53 63 73 34 66)(10 54 64 74 35 67)(11 55 57 75 36 68)(12 56 58 76 37 69)(13 49 59 77 38 70)(14 50 60 78 39 71)(15 51 61 79 40 72)(16 52 62 80 33 65)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 89)(8 90)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 88)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 37)(34 38)(35 39)(36 40)(49 53)(50 54)(51 55)(52 56)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(41 75 45 79)(42 74 46 78)(43 73 47 77)(44 80 48 76)(49 93 53 89)(50 92 54 96)(51 91 55 95)(52 90 56 94)(57 82 61 86)(58 81 62 85)(59 88 63 84)(60 87 64 83)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C4○D4 Q16 D12 C4○D8 Q8⋊3S3 S3×D4 D8⋊3S3 S3×Q16 kernel D6⋊2Q16 C6.SD16 C3×C2.D8 C4.D12 S3×C2×C8 C2×Dic12 C2.D8 C24 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C12 D6 C8 C6 C4 C22 C2 C2 # reps 1 2 1 2 1 1 1 2 1 1 2 1 2 4 4 4 1 1 2 2

Matrix representation of D62Q16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 1
,
 1 0 0 0 0 72 0 0 0 0 72 1 0 0 0 1
,
 51 0 0 0 0 63 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 72 0 0 0 0 0 59 7 0 0 66 14
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,1,1],[51,0,0,0,0,63,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,0,0,0,0,0,59,66,0,0,7,14] >;

D62Q16 in GAP, Magma, Sage, TeX

D_6\rtimes_2Q_{16}
% in TeX

G:=Group("D6:2Q16");
// GroupNames label

G:=SmallGroup(192,446);
// by ID

G=gap.SmallGroup(192,446);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,438,102,6278]);
// Polycyclic

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

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