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## G = Dic6⋊17D4order 192 = 26·3

### 5th semidirect product of Dic6 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊17D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C22×Dic6 — Dic6⋊17D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊17D4
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for Dic617D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >

Subgroups: 432 in 158 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C6 [×3], C6 [×3], C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×10], C23, C23, Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×7], C3⋊C8 [×2], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C2×C3⋊C8 [×2], D4.S3 [×4], C3×C22⋊C4, C3×C4⋊C4, C2×Dic6 [×2], C2×Dic6 [×5], C22×Dic3, C22×C12, C6×D4, C6×D4, Q8⋊D4, C6.SD16 [×2], C12.55D4, C2×D4.S3 [×2], C3×C4⋊D4, C22×Dic6, Dic617D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×2], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C2×SD16, C8.C22, D4.S3 [×2], S3×D4 [×2], C2×C3⋊D4, Q8⋊D4, C2×D4.S3, C232D6, Q8.14D6, Dic617D4

Smallest permutation representation of Dic617D4
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 74 7 80)(2 73 8 79)(3 84 9 78)(4 83 10 77)(5 82 11 76)(6 81 12 75)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)(49 89 55 95)(50 88 56 94)(51 87 57 93)(52 86 58 92)(53 85 59 91)(54 96 60 90)
(1 54 34 40)(2 49 35 47)(3 56 36 42)(4 51 25 37)(5 58 26 44)(6 53 27 39)(7 60 28 46)(8 55 29 41)(9 50 30 48)(10 57 31 43)(11 52 32 38)(12 59 33 45)(13 77 96 63)(14 84 85 70)(15 79 86 65)(16 74 87 72)(17 81 88 67)(18 76 89 62)(19 83 90 69)(20 78 91 64)(21 73 92 71)(22 80 93 66)(23 75 94 61)(24 82 95 68)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 57)(26 58)(27 59)(28 60)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 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), (1,74,7,80)(2,73,8,79)(3,84,9,78)(4,83,10,77)(5,82,11,76)(6,81,12,75)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)(49,89,55,95)(50,88,56,94)(51,87,57,93)(52,86,58,92)(53,85,59,91)(54,96,60,90), (1,54,34,40)(2,49,35,47)(3,56,36,42)(4,51,25,37)(5,58,26,44)(6,53,27,39)(7,60,28,46)(8,55,29,41)(9,50,30,48)(10,57,31,43)(11,52,32,38)(12,59,33,45)(13,77,96,63)(14,84,85,70)(15,79,86,65)(16,74,87,72)(17,81,88,67)(18,76,89,62)(19,83,90,69)(20,78,91,64)(21,73,92,71)(22,80,93,66)(23,75,94,61)(24,82,95,68), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,37)(11,38)(12,39)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,57)(26,58)(27,59)(28,60)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)>;

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

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

33 conjugacy classes

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

33 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 D6 SD16 C3⋊D4 C3⋊D4 C8.C22 S3×D4 D4.S3 Q8.14D6 kernel Dic6⋊17D4 C6.SD16 C12.55D4 C2×D4.S3 C3×C4⋊D4 C22×Dic6 C4⋊D4 Dic6 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 4 1 1 1 1 1 4 2 2 1 2 2 2

Matrix representation of Dic617D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 65 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 2 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 67 0 0 0 0 67 67
,
 1 71 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1
,
 1 71 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,65,0,0,0,0,0,0,9,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[1,0,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

Dic617D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{17}D_4
% in TeX

G:=Group("Dic6:17D4");
// GroupNames label

G:=SmallGroup(192,599);
// by ID

G=gap.SmallGroup(192,599);
# by ID

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

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

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