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## G = Dic6⋊D4order 192 = 26·3

### 6th semidirect product of Dic6 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic6⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×D4⋊2S3 — Dic6⋊D4
 Lower central C3 — C6 — C2×C12 — Dic6⋊D4
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for Dic6⋊D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=cac-1=a-1, dad=a5, cbc-1=a9b, dbd=a6b, dcd=c-1 >

Subgroups: 504 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C2×C3⋊C8, C4⋊Dic3, D4.S3, C6.D4, C2×C24, C3×D8, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, D4⋊D4, C2.Dic12, D6⋊C8, D4⋊Dic3, C2×D4.S3, D63D4, C6×D8, C2×D42S3, Dic6⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8⋊C22, S3×D4, C2×C3⋊D4, D4⋊D4, D8⋊S3, D83S3, C232D6, Dic6⋊D4

Smallest permutation representation of Dic6⋊D4
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 55 7 49)(2 54 8 60)(3 53 9 59)(4 52 10 58)(5 51 11 57)(6 50 12 56)(13 90 19 96)(14 89 20 95)(15 88 21 94)(16 87 22 93)(17 86 23 92)(18 85 24 91)(25 70 31 64)(26 69 32 63)(27 68 33 62)(28 67 34 61)(29 66 35 72)(30 65 36 71)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)
(1 77 94 68)(2 76 95 67)(3 75 96 66)(4 74 85 65)(5 73 86 64)(6 84 87 63)(7 83 88 62)(8 82 89 61)(9 81 90 72)(10 80 91 71)(11 79 92 70)(12 78 93 69)(13 26 53 45)(14 25 54 44)(15 36 55 43)(16 35 56 42)(17 34 57 41)(18 33 58 40)(19 32 59 39)(20 31 60 38)(21 30 49 37)(22 29 50 48)(23 28 51 47)(24 27 52 46)
(1 94)(2 87)(3 92)(4 85)(5 90)(6 95)(7 88)(8 93)(9 86)(10 91)(11 96)(12 89)(13 51)(14 56)(15 49)(16 54)(17 59)(18 52)(19 57)(20 50)(21 55)(22 60)(23 53)(24 58)(25 35)(26 28)(27 33)(29 31)(30 36)(32 34)(37 43)(38 48)(39 41)(40 46)(42 44)(45 47)(61 69)(63 67)(64 72)(66 70)(73 81)(75 79)(76 84)(78 82)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 C3⋊D4 C4○D8 C8⋊C22 S3×D4 S3×D4 D8⋊S3 D8⋊3S3 kernel Dic6⋊D4 C2.Dic12 D6⋊C8 D4⋊Dic3 C2×D4.S3 D6⋊3D4 C6×D8 C2×D4⋊2S3 C2×D8 Dic6 C2×Dic3 C3×D4 C22×S3 C2×C8 C2×D4 D4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 2 1 1 2 4 4 1 1 1 2 2

Matrix representation of Dic6⋊D4 in GL4(𝔽73) generated by

 46 0 0 0 0 27 0 0 0 0 1 72 0 0 1 0
,
 0 22 0 0 63 0 0 0 0 0 0 72 0 0 72 0
,
 0 1 0 0 72 0 0 0 0 0 13 30 0 0 43 60
,
 72 0 0 0 0 1 0 0 0 0 0 72 0 0 72 0
G:=sub<GL(4,GF(73))| [46,0,0,0,0,27,0,0,0,0,1,1,0,0,72,0],[0,63,0,0,22,0,0,0,0,0,0,72,0,0,72,0],[0,72,0,0,1,0,0,0,0,0,13,43,0,0,30,60],[72,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0] >;

Dic6⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_4
% in TeX

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

G:=SmallGroup(192,717);
// by ID

G=gap.SmallGroup(192,717);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,254,219,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^9*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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