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## G = C3×Dic3⋊C8order 288 = 25·32

### Direct product of C3 and Dic3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Dic3⋊C8
G = < a,b,c,d | a3=b6=d8=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 154 in 87 conjugacy classes, 50 normal (46 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], Dic3, C12 [×4], C12 [×5], C2×C6 [×2], C2×C6, C42, C2×C8, C2×C8, C3×C6 [×3], C3⋊C8, C24 [×5], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4⋊C8, C3×Dic3 [×2], C3×Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24 [×2], C2×C24 [×2], C3×C3⋊C8, C3×C24, C6×Dic3 [×2], C6×C12, Dic3⋊C8, C3×C4⋊C8, C6×C3⋊C8, Dic3×C12, C6×C24, C3×Dic3⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, D4, Q8, C12 [×2], D6, C2×C6, C4⋊C4, C2×C8, M4(2), C3×S3, C24 [×2], Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4⋊C8, S3×C6, S3×C8, C8⋊S3, Dic3⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×Dic6, S3×C12, C3×C3⋊D4, Dic3⋊C8, C3×C4⋊C8, S3×C24, C3×C8⋊S3, C3×Dic3⋊C4, C3×Dic3⋊C8

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I ··· 12T 12U ··· 12AB 24A ··· 24AF 24AG ··· 24AN order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 1 1 2 2 2 1 1 1 1 6 6 6 6 1 ··· 1 2 ··· 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C8 C12 C24 S3 D4 Q8 D6 M4(2) C3×S3 Dic6 C3⋊D4 C3×D4 C3×Q8 C4×S3 S3×C6 S3×C8 C8⋊S3 C3×M4(2) C3×Dic6 C3×C3⋊D4 S3×C12 S3×C24 C3×C8⋊S3 kernel C3×Dic3⋊C8 C6×C3⋊C8 Dic3×C12 C6×C24 Dic3⋊C8 C6×Dic3 C2×C3⋊C8 C4×Dic3 C2×C24 C3×Dic3 C2×Dic3 Dic3 C2×C24 C3×C12 C3×C12 C2×C12 C3×C6 C2×C8 C12 C12 C12 C12 C2×C6 C2×C4 C6 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 8 16 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8

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

 64 0 0 0 0 64 0 0 0 0 8 0 0 0 0 8
,
 64 0 0 0 51 8 0 0 0 0 65 0 0 0 51 9
,
 52 66 0 0 42 21 0 0 0 0 30 10 0 0 34 43
,
 63 0 0 0 0 63 0 0 0 0 22 0 0 0 14 51
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[64,51,0,0,0,8,0,0,0,0,65,51,0,0,0,9],[52,42,0,0,66,21,0,0,0,0,30,34,0,0,10,43],[63,0,0,0,0,63,0,0,0,0,22,14,0,0,0,51] >;

C3×Dic3⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3:C8");
// GroupNames label

G:=SmallGroup(288,248);
// by ID

G=gap.SmallGroup(288,248);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,136,9414]);
// Polycyclic

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

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