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## G = Dic6⋊9D4order 192 = 26·3

### 2nd semidirect product of Dic6 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊9D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C4×Dic6 — Dic6⋊9D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊9D4
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for Dic69D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a9b, dcd=c-1 >

Subgroups: 352 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C4×C12, C2×Dic6, C6×D4, C6×D4, C4⋊SD16, C12⋊C8, D4⋊Dic3, C4×Dic6, C2×D4.S3, C3×C41D4, Dic69D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, D4.S3, S3×D4, D42S3, C2×C3⋊D4, C4⋊SD16, D126C22, C2×D4.S3, D63D4, Dic69D4

Smallest permutation representation of Dic69D4
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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 71 19 65)(14 70 20 64)(15 69 21 63)(16 68 22 62)(17 67 23 61)(18 66 24 72)(37 80 43 74)(38 79 44 73)(39 78 45 84)(40 77 46 83)(41 76 47 82)(42 75 48 81)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)
(1 13 52 82)(2 14 53 83)(3 15 54 84)(4 16 55 73)(5 17 56 74)(6 18 57 75)(7 19 58 76)(8 20 59 77)(9 21 60 78)(10 22 49 79)(11 23 50 80)(12 24 51 81)(25 63 91 45)(26 64 92 46)(27 65 93 47)(28 66 94 48)(29 67 95 37)(30 68 96 38)(31 69 85 39)(32 70 86 40)(33 71 87 41)(34 72 88 42)(35 61 89 43)(36 62 90 44)
(2 8)(4 10)(6 12)(13 82)(14 77)(15 84)(16 79)(17 74)(18 81)(19 76)(20 83)(21 78)(22 73)(23 80)(24 75)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 70)(38 65)(39 72)(40 67)(41 62)(42 69)(43 64)(44 71)(45 66)(46 61)(47 68)(48 63)(49 55)(51 57)(53 59)(85 88)(86 95)(87 90)(89 92)(91 94)(93 96)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C4○D4 C3⋊D4 C8⋊C22 D4.S3 S3×D4 D4⋊2S3 D12⋊6C22 kernel Dic6⋊9D4 C12⋊C8 D4⋊Dic3 C4×Dic6 C2×D4.S3 C3×C4⋊1D4 C4⋊1D4 Dic6 C2×C12 C42 C2×D4 C12 C12 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 2 1 2 1 1 2 2 1 2 4 2 4 1 2 1 1 2

Matrix representation of Dic69D4 in GL6(𝔽73)

 9 0 0 0 0 0 0 65 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 6 67 0 0 0 0 67 67
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

Dic69D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_9D_4
% in TeX

G:=Group("Dic6:9D4");
// GroupNames label

G:=SmallGroup(192,634);
// by ID

G=gap.SmallGroup(192,634);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations

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