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G = D8⋊1Dic3order 192 = 26·3

1st semidirect product of D8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D8⋊1Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24⋊1C4 — D8⋊1Dic3
 Lower central C3 — C6 — C12 — C24 — D8⋊1Dic3
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×D8

Generators and relations for D81Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 216 in 66 conjugacy classes, 31 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C2×C6, C2×C6, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, C24, C2×Dic3, C2×C12, C3×D4, C22×C6, C2.D8, C2×C16, C2×D8, C3⋊C16, C4⋊Dic3, C2×C24, C3×D8, C3×D8, C6×D4, C2.D16, C2×C3⋊C16, C241C4, C6×D8, D81Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, D4⋊C4, D16, SD32, D4⋊S3, D4.S3, C6.D4, C2.D16, C3⋊D16, D8.S3, D4⋊Dic3, D81Dic3

Smallest permutation representation of D81Dic3
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 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 84)(10 83)(11 82)(12 81)(13 88)(14 87)(15 86)(16 85)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 44)(26 43)(27 42)(28 41)(29 48)(30 47)(31 46)(32 45)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 56)(40 55)(65 89)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)
(1 49 32 62 39 46)(2 50 25 63 40 47)(3 51 26 64 33 48)(4 52 27 57 34 41)(5 53 28 58 35 42)(6 54 29 59 36 43)(7 55 30 60 37 44)(8 56 31 61 38 45)(9 72 22 82 96 73)(10 65 23 83 89 74)(11 66 24 84 90 75)(12 67 17 85 91 76)(13 68 18 86 92 77)(14 69 19 87 93 78)(15 70 20 88 94 79)(16 71 21 81 95 80)
(1 11 62 84)(2 10 63 83)(3 9 64 82)(4 16 57 81)(5 15 58 88)(6 14 59 87)(7 13 60 86)(8 12 61 85)(17 56 76 38)(18 55 77 37)(19 54 78 36)(20 53 79 35)(21 52 80 34)(22 51 73 33)(23 50 74 40)(24 49 75 39)(25 89 47 65)(26 96 48 72)(27 95 41 71)(28 94 42 70)(29 93 43 69)(30 92 44 68)(31 91 45 67)(32 90 46 66)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 16A ··· 16H 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 16 ··· 16 24 24 24 24 size 1 1 1 1 8 8 2 2 2 24 24 2 2 2 8 8 8 8 2 2 2 2 4 4 6 ··· 6 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - + + - image C1 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 SD16 D8 C3⋊D4 C3⋊D4 D16 SD32 D4.S3 D4⋊S3 C3⋊D16 D8.S3 kernel D8⋊1Dic3 C2×C3⋊C16 C24⋊1C4 C6×D8 C3×D8 C2×D8 C24 C2×C12 C2×C8 D8 C12 C2×C6 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 1 1 2 2

Matrix representation of D81Dic3 in GL6(𝔽97)

 90 7 0 0 0 0 90 90 0 0 0 0 0 0 90 7 0 0 0 0 90 90 0 0 0 0 0 0 96 0 0 0 0 0 0 96
,
 90 7 0 0 0 0 7 7 0 0 0 0 0 0 7 90 0 0 0 0 90 90 0 0 0 0 0 0 96 0 0 0 0 0 31 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 0 0 0 0 0 0 96 0 0 0 0 0 0 61 0 0 0 0 0 82 35
,
 26 95 0 0 0 0 95 71 0 0 0 0 0 0 10 44 0 0 0 0 44 87 0 0 0 0 0 0 31 2 0 0 0 0 5 66

G:=sub<GL(6,GF(97))| [90,90,0,0,0,0,7,90,0,0,0,0,0,0,90,90,0,0,0,0,7,90,0,0,0,0,0,0,96,0,0,0,0,0,0,96],[90,7,0,0,0,0,7,7,0,0,0,0,0,0,7,90,0,0,0,0,90,90,0,0,0,0,0,0,96,31,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,61,82,0,0,0,0,0,35],[26,95,0,0,0,0,95,71,0,0,0,0,0,0,10,44,0,0,0,0,44,87,0,0,0,0,0,0,31,5,0,0,0,0,2,66] >;

D81Dic3 in GAP, Magma, Sage, TeX

D_8\rtimes_1{\rm Dic}_3
% in TeX

G:=Group("D8:1Dic3");
// GroupNames label

G:=SmallGroup(192,121);
// by ID

G=gap.SmallGroup(192,121);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,675,346,192,1684,851,102,6278]);
// Polycyclic

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

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