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## G = C3⋊C8⋊Dic3order 288 = 25·32

### 3rd semidirect product of C3⋊C8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3⋊C8⋊Dic3
G = < a,b,c,d | a3=b8=c6=1, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >

Subgroups: 210 in 91 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C8⋊C4, C3×Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8, C2×C3⋊C8 [×3], C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8 [×2], C324C8 [×2], C6×Dic3 [×2], C6×C12, C42.S3, C24⋊C4, C6×C3⋊C8, Dic3×C12, C2×C324C8, C3⋊C8⋊Dic3
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×2], C2×C4 [×3], Dic3 [×4], D6 [×2], C42, M4(2) [×2], C4×S3 [×4], C2×Dic3 [×2], C8⋊C4, S32, C8⋊S3 [×2], C4.Dic3 [×2], C4×Dic3 [×2], S3×Dic3 [×2], C6.D6, C42.S3, C24⋊C4, D6.Dic3 [×2], Dic32, C3⋊C8⋊Dic3

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T 24A ··· 24H order 1 2 2 2 3 3 3 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 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 6 6 6 6 2 ··· 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 4 4 4 6 ··· 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - - + + - + - image C1 C2 C2 C2 C4 C4 C4 S3 S3 Dic3 Dic3 D6 M4(2) C4×S3 C4×S3 C8⋊S3 C4.Dic3 S32 S3×Dic3 C6.D6 S3×Dic3 D6.Dic3 kernel C3⋊C8⋊Dic3 C6×C3⋊C8 Dic3×C12 C2×C32⋊4C8 C3×C3⋊C8 C32⋊4C8 C6×Dic3 C2×C3⋊C8 C4×Dic3 C3⋊C8 C2×Dic3 C2×C12 C3×C6 C12 C2×C6 C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 1 1 1 4 4 4 1 1 2 2 2 4 6 2 8 8 1 1 1 1 4

Matrix representation of C3⋊C8⋊Dic3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 0 1 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 70 0 0 0 0 8 13 0 0 0 0 0 0 27 0 0 0 0 0 46 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,27,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,8,0,0,0,0,70,13,0,0,0,0,0,0,27,46,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3⋊C8⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3:C8:Dic3");
// GroupNames label

G:=SmallGroup(288,202);
// by ID

G=gap.SmallGroup(288,202);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,253,36,100,1356,9414]);
// Polycyclic

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

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