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## G = Dic3⋊4D8order 192 = 26·3

### 1st semidirect product of Dic3 and D8 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Dic3⋊4D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — D4×Dic3 — Dic3⋊4D8
 Lower central C3 — C6 — C12 — Dic3⋊4D8
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic34D8
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 408 in 134 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C4×C8, D4⋊C4, D4⋊C4, C2.D8, C4×D4, C2×D8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, D4⋊S3, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C22×Dic3, C6×D4, C4×D8, C6.Q16, C8×Dic3, C2.D24, C3×D4⋊C4, Dic35D4, C2×D4⋊S3, D4×Dic3, Dic34D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, D8, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×D8, C4○D8, S3×C2×C4, S3×D4, D42S3, C4×D8, Dic34D4, S3×D8, Q8.7D6, Dic34D8

Smallest permutation representation of Dic34D8
On 96 points
Generators in S96
(1 40 24 65 84 91)(2 92 85 66 17 33)(3 34 18 67 86 93)(4 94 87 68 19 35)(5 36 20 69 88 95)(6 96 81 70 21 37)(7 38 22 71 82 89)(8 90 83 72 23 39)(9 55 59 77 28 45)(10 46 29 78 60 56)(11 49 61 79 30 47)(12 48 31 80 62 50)(13 51 63 73 32 41)(14 42 25 74 64 52)(15 53 57 75 26 43)(16 44 27 76 58 54)
(1 52 65 25)(2 53 66 26)(3 54 67 27)(4 55 68 28)(5 56 69 29)(6 49 70 30)(7 50 71 31)(8 51 72 32)(9 19 77 94)(10 20 78 95)(11 21 79 96)(12 22 80 89)(13 23 73 90)(14 24 74 91)(15 17 75 92)(16 18 76 93)(33 57 85 43)(34 58 86 44)(35 59 87 45)(36 60 88 46)(37 61 81 47)(38 62 82 48)(39 63 83 41)(40 64 84 42)
(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 4)(2 3)(5 8)(6 7)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 88)(24 87)(25 28)(26 27)(29 32)(30 31)(33 93)(34 92)(35 91)(36 90)(37 89)(38 96)(39 95)(40 94)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 80)(48 79)(49 50)(51 56)(52 55)(53 54)(65 68)(66 67)(69 72)(70 71)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 D8 C4○D4 C4×S3 C4○D8 D4⋊2S3 S3×D4 S3×D8 Q8.7D6 kernel Dic3⋊4D8 C6.Q16 C8×Dic3 C2.D24 C3×D4⋊C4 Dic3⋊5D4 C2×D4⋊S3 D4×Dic3 D4⋊S3 D4⋊C4 C2×Dic3 C4⋊C4 C2×C8 C2×D4 Dic3 C12 D4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 2 4 4 1 1 2 2

Matrix representation of Dic34D8 in GL4(𝔽73) generated by

 1 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 0 46 0 0 46 0 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 0 32 0 0 57 32
,
 72 0 0 0 0 72 0 0 0 0 0 41 0 0 57 0
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,46,0,0,46,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,0,57,0,0,32,32],[72,0,0,0,0,72,0,0,0,0,0,57,0,0,41,0] >;

Dic34D8 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_4D_8
% in TeX

G:=Group("Dic3:4D8");
// GroupNames label

G:=SmallGroup(192,315);
// by ID

G=gap.SmallGroup(192,315);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,135,100,570,297,136,6278]);
// Polycyclic

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

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