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G = Dic3⋊C8order 96 = 25·3

The semidirect product of Dic3 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3⋊C8
G = < a,b,c | a6=c8=1, b2=a3, bab-1=a-1, ac=ca, cbc-1=a3b >

Smallest permutation representation of Dic3⋊C8
Regular action on 96 points
Generators in S96
(1 53 62 23 43 77)(2 54 63 24 44 78)(3 55 64 17 45 79)(4 56 57 18 46 80)(5 49 58 19 47 73)(6 50 59 20 48 74)(7 51 60 21 41 75)(8 52 61 22 42 76)(9 31 96 66 86 33)(10 32 89 67 87 34)(11 25 90 68 88 35)(12 26 91 69 81 36)(13 27 92 70 82 37)(14 28 93 71 83 38)(15 29 94 72 84 39)(16 30 95 65 85 40)
(1 11 23 68)(2 69 24 12)(3 13 17 70)(4 71 18 14)(5 15 19 72)(6 65 20 16)(7 9 21 66)(8 67 22 10)(25 62 88 77)(26 78 81 63)(27 64 82 79)(28 80 83 57)(29 58 84 73)(30 74 85 59)(31 60 86 75)(32 76 87 61)(33 41 96 51)(34 52 89 42)(35 43 90 53)(36 54 91 44)(37 45 92 55)(38 56 93 46)(39 47 94 49)(40 50 95 48)
(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)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C8 S3 D4 Q8 D6 M4(2) Dic6 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel Dic3⋊C8 C2×C3⋊C8 C4×Dic3 C2×C24 C2×Dic3 Dic3 C2×C8 C12 C12 C2×C4 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 8 1 1 1 1 2 2 2 2 4 4

Matrix representation of Dic3⋊C8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 72 0 0 1 0
,
 23 32 0 0 61 50 0 0 0 0 8 26 0 0 34 65
,
 0 1 0 0 46 0 0 0 0 0 43 60 0 0 13 30
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[23,61,0,0,32,50,0,0,0,0,8,34,0,0,26,65],[0,46,0,0,1,0,0,0,0,0,43,13,0,0,60,30] >;

Dic3⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes C_8
% in TeX

G:=Group("Dic3:C8");
// GroupNames label

G:=SmallGroup(96,21);
// by ID

G=gap.SmallGroup(96,21);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,86,2309]);
// Polycyclic

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

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