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## G = Dic3×C3⋊C8order 288 = 25·32

### Direct product of Dic3 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — Dic3×C3⋊C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — Dic3×C12 — Dic3×C3⋊C8
 Lower central C32 — Dic3×C3⋊C8
 Upper central C1 — C2×C4

Generators and relations for Dic3×C3⋊C8
G = < a,b,c,d | a6=c3=d8=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 210 in 99 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4×C8, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, C324C8, C6×Dic3, C6×C12, C4×C3⋊C8, C8×Dic3, C6×C3⋊C8, Dic3×C12, C2×C324C8, Dic3×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, C3⋊C8, C4×S3, C2×Dic3, C4×C8, S32, S3×C8, C2×C3⋊C8, C4×Dic3, S3×Dic3, C6.D6, C4×C3⋊C8, C8×Dic3, S3×C3⋊C8, Dic32, Dic3×C3⋊C8

Smallest permutation representation of Dic3×C3⋊C8
On 96 points
Generators in S96
(1 75 64 54 88 21)(2 76 57 55 81 22)(3 77 58 56 82 23)(4 78 59 49 83 24)(5 79 60 50 84 17)(6 80 61 51 85 18)(7 73 62 52 86 19)(8 74 63 53 87 20)(9 25 37 95 71 46)(10 26 38 96 72 47)(11 27 39 89 65 48)(12 28 40 90 66 41)(13 29 33 91 67 42)(14 30 34 92 68 43)(15 31 35 93 69 44)(16 32 36 94 70 45)
(1 10 54 96)(2 11 55 89)(3 12 56 90)(4 13 49 91)(5 14 50 92)(6 15 51 93)(7 16 52 94)(8 9 53 95)(17 30 60 68)(18 31 61 69)(19 32 62 70)(20 25 63 71)(21 26 64 72)(22 27 57 65)(23 28 58 66)(24 29 59 67)(33 78 42 83)(34 79 43 84)(35 80 44 85)(36 73 45 86)(37 74 46 87)(38 75 47 88)(39 76 48 81)(40 77 41 82)
(1 88 64)(2 57 81)(3 82 58)(4 59 83)(5 84 60)(6 61 85)(7 86 62)(8 63 87)(9 71 37)(10 38 72)(11 65 39)(12 40 66)(13 67 33)(14 34 68)(15 69 35)(16 36 70)(17 50 79)(18 80 51)(19 52 73)(20 74 53)(21 54 75)(22 76 55)(23 56 77)(24 78 49)(25 46 95)(26 96 47)(27 48 89)(28 90 41)(29 42 91)(30 92 43)(31 44 93)(32 94 45)
(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)| (1,75,64,54,88,21)(2,76,57,55,81,22)(3,77,58,56,82,23)(4,78,59,49,83,24)(5,79,60,50,84,17)(6,80,61,51,85,18)(7,73,62,52,86,19)(8,74,63,53,87,20)(9,25,37,95,71,46)(10,26,38,96,72,47)(11,27,39,89,65,48)(12,28,40,90,66,41)(13,29,33,91,67,42)(14,30,34,92,68,43)(15,31,35,93,69,44)(16,32,36,94,70,45), (1,10,54,96)(2,11,55,89)(3,12,56,90)(4,13,49,91)(5,14,50,92)(6,15,51,93)(7,16,52,94)(8,9,53,95)(17,30,60,68)(18,31,61,69)(19,32,62,70)(20,25,63,71)(21,26,64,72)(22,27,57,65)(23,28,58,66)(24,29,59,67)(33,78,42,83)(34,79,43,84)(35,80,44,85)(36,73,45,86)(37,74,46,87)(38,75,47,88)(39,76,48,81)(40,77,41,82), (1,88,64)(2,57,81)(3,82,58)(4,59,83)(5,84,60)(6,61,85)(7,86,62)(8,63,87)(9,71,37)(10,38,72)(11,65,39)(12,40,66)(13,67,33)(14,34,68)(15,69,35)(16,36,70)(17,50,79)(18,80,51)(19,52,73)(20,74,53)(21,54,75)(22,76,55)(23,56,77)(24,78,49)(25,46,95)(26,96,47)(27,48,89)(28,90,41)(29,42,91)(30,92,43)(31,44,93)(32,94,45), (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( (1,75,64,54,88,21)(2,76,57,55,81,22)(3,77,58,56,82,23)(4,78,59,49,83,24)(5,79,60,50,84,17)(6,80,61,51,85,18)(7,73,62,52,86,19)(8,74,63,53,87,20)(9,25,37,95,71,46)(10,26,38,96,72,47)(11,27,39,89,65,48)(12,28,40,90,66,41)(13,29,33,91,67,42)(14,30,34,92,68,43)(15,31,35,93,69,44)(16,32,36,94,70,45), (1,10,54,96)(2,11,55,89)(3,12,56,90)(4,13,49,91)(5,14,50,92)(6,15,51,93)(7,16,52,94)(8,9,53,95)(17,30,60,68)(18,31,61,69)(19,32,62,70)(20,25,63,71)(21,26,64,72)(22,27,57,65)(23,28,58,66)(24,29,59,67)(33,78,42,83)(34,79,43,84)(35,80,44,85)(36,73,45,86)(37,74,46,87)(38,75,47,88)(39,76,48,81)(40,77,41,82), (1,88,64)(2,57,81)(3,82,58)(4,59,83)(5,84,60)(6,61,85)(7,86,62)(8,63,87)(9,71,37)(10,38,72)(11,65,39)(12,40,66)(13,67,33)(14,34,68)(15,69,35)(16,36,70)(17,50,79)(18,80,51)(19,52,73)(20,74,53)(21,54,75)(22,76,55)(23,56,77)(24,78,49)(25,46,95)(26,96,47)(27,48,89)(28,90,41)(29,42,91)(30,92,43)(31,44,93)(32,94,45), (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([[(1,75,64,54,88,21),(2,76,57,55,81,22),(3,77,58,56,82,23),(4,78,59,49,83,24),(5,79,60,50,84,17),(6,80,61,51,85,18),(7,73,62,52,86,19),(8,74,63,53,87,20),(9,25,37,95,71,46),(10,26,38,96,72,47),(11,27,39,89,65,48),(12,28,40,90,66,41),(13,29,33,91,67,42),(14,30,34,92,68,43),(15,31,35,93,69,44),(16,32,36,94,70,45)], [(1,10,54,96),(2,11,55,89),(3,12,56,90),(4,13,49,91),(5,14,50,92),(6,15,51,93),(7,16,52,94),(8,9,53,95),(17,30,60,68),(18,31,61,69),(19,32,62,70),(20,25,63,71),(21,26,64,72),(22,27,57,65),(23,28,58,66),(24,29,59,67),(33,78,42,83),(34,79,43,84),(35,80,44,85),(36,73,45,86),(37,74,46,87),(38,75,47,88),(39,76,48,81),(40,77,41,82)], [(1,88,64),(2,57,81),(3,82,58),(4,59,83),(5,84,60),(6,61,85),(7,86,62),(8,63,87),(9,71,37),(10,38,72),(11,65,39),(12,40,66),(13,67,33),(14,34,68),(15,69,35),(16,36,70),(17,50,79),(18,80,51),(19,52,73),(20,74,53),(21,54,75),(22,76,55),(23,56,77),(24,78,49),(25,46,95),(26,96,47),(27,48,89),(28,90,41),(29,42,91),(30,92,43),(31,44,93),(32,94,45)], [(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)]])

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G 6H 6I 8A ··· 8H 8I ··· 8P 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 3 ··· 3 2 ··· 2 4 4 4 3 ··· 3 9 ··· 9 2 ··· 2 4 4 4 4 6 ··· 6 6 ··· 6

72 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 C4 C4 C4 C8 S3 S3 Dic3 Dic3 D6 C3⋊C8 C4×S3 C4×S3 S3×C8 S32 S3×Dic3 C6.D6 S3×Dic3 S3×C3⋊C8 kernel Dic3×C3⋊C8 C6×C3⋊C8 Dic3×C12 C2×C32⋊4C8 C3×C3⋊C8 C32⋊4C8 C6×Dic3 C3×Dic3 C2×C3⋊C8 C4×Dic3 C3⋊C8 C2×Dic3 C2×C12 Dic3 C12 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 1 1 1 4 4 4 16 1 1 2 2 2 8 6 2 8 1 1 1 1 4

Matrix representation of Dic3×C3⋊C8 in GL4(𝔽73) generated by

 0 1 0 0 72 1 0 0 0 0 72 0 0 0 0 72
,
 0 27 0 0 27 0 0 0 0 0 46 0 0 0 0 46
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 51 0 0 0 0 51 0 0 0 0 27 46 0 0 0 46
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[0,27,0,0,27,0,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[51,0,0,0,0,51,0,0,0,0,27,0,0,0,46,46] >;

Dic3×C3⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\rtimes C_8
% in TeX

G:=Group("Dic3xC3:C8");
// GroupNames label

G:=SmallGroup(288,200);
// by ID

G=gap.SmallGroup(288,200);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,36,100,1356,9414]);
// Polycyclic

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

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