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## G = D8⋊Dic3order 192 = 26·3

### 3rd semidirect product of D8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D8⋊Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 360 in 132 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×D8, C22×Dic3, C6×D4, D8⋊C4, C24⋊C4, C8⋊Dic3, D4⋊Dic3, D4×Dic3, C6×D8, D8⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C8⋊C22, S3×D4, D42S3, C22×Dic3, D8⋊C4, D8⋊S3, D4×Dic3, D8⋊Dic3

Smallest permutation representation of D8⋊Dic3
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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 24)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 56)(54 55)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)(85 88)(86 87)(89 92)(90 91)(93 96)(94 95)
(1 62 51 77 43 24)(2 63 52 78 44 17)(3 64 53 79 45 18)(4 57 54 80 46 19)(5 58 55 73 47 20)(6 59 56 74 48 21)(7 60 49 75 41 22)(8 61 50 76 42 23)(9 30 35 67 91 87)(10 31 36 68 92 88)(11 32 37 69 93 81)(12 25 38 70 94 82)(13 26 39 71 95 83)(14 27 40 72 96 84)(15 28 33 65 89 85)(16 29 34 66 90 86)
(1 11 77 69)(2 16 78 66)(3 13 79 71)(4 10 80 68)(5 15 73 65)(6 12 74 70)(7 9 75 67)(8 14 76 72)(17 29 52 90)(18 26 53 95)(19 31 54 92)(20 28 55 89)(21 25 56 94)(22 30 49 91)(23 27 50 96)(24 32 51 93)(33 58 85 47)(34 63 86 44)(35 60 87 41)(36 57 88 46)(37 62 81 43)(38 59 82 48)(39 64 83 45)(40 61 84 42)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 C4○D4 C8⋊C22 D4⋊2S3 S3×D4 D8⋊S3 kernel D8⋊Dic3 C24⋊C4 C8⋊Dic3 D4⋊Dic3 D4×Dic3 C6×D8 C3×D8 C2×D8 C2×Dic3 C2×C8 D8 C2×D4 C12 C6 C4 C22 C2 # reps 1 1 1 2 2 1 8 1 2 1 4 2 2 2 1 1 4

Matrix representation of D8⋊Dic3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 62 0 0 0 0 0 11 0 0 53 53 62 22 0 0 0 53 51 11
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 11 62 0 0 0 0 0 11 0 0 20 20 0 0 0 0 0 20 0 0
,
 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 1 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 43 43 0 0 0 0 13 30 0 0 0 0 0 0 31 3 41 47 0 0 45 42 47 58 0 0 14 39 3 42 0 0 39 25 45 70

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,53,0,0,0,0,0,53,53,0,0,11,0,62,51,0,0,62,11,22,11],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,20,0,0,0,0,0,20,20,0,0,11,0,0,0,0,0,62,11,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[43,13,0,0,0,0,43,30,0,0,0,0,0,0,31,45,14,39,0,0,3,42,39,25,0,0,41,47,3,45,0,0,47,58,42,70] >;

D8⋊Dic3 in GAP, Magma, Sage, TeX

D_8\rtimes {\rm Dic}_3
% in TeX

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

G:=SmallGroup(192,711);
// by ID

G=gap.SmallGroup(192,711);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,219,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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