metabelian, supersoluble, monomial
Aliases: C3⋊C8⋊3Dic3, C2.3Dic32, C12.45(C4×S3), C32⋊4C8⋊6C4, C3⋊2(C24⋊C4), (C3×C6).8C42, C6.3(C4×Dic3), C32⋊3(C8⋊C4), (C2×C12).293D6, C62.23(C2×C4), (C6×Dic3).3C4, (C4×Dic3).6S3, C4.20(S3×Dic3), C6.11(C8⋊S3), (C3×C6).3M4(2), C12.31(C2×Dic3), C6.1(C4.Dic3), C22.9(S3×Dic3), (C6×C12).198C22, C3⋊1(C42.S3), (Dic3×C12).15C2, (C2×Dic3).2Dic3, C4.14(C6.D6), C2.1(D6.Dic3), (C3×C3⋊C8)⋊6C4, (C2×C3⋊C8).7S3, (C6×C3⋊C8).18C2, (C2×C4).126S32, (C2×C6).64(C4×S3), (C3×C12).83(C2×C4), (C2×C6).13(C2×Dic3), (C2×C32⋊4C8).13C2, SmallGroup(288,202)
Series: Derived ►Chief ►Lower central ►Upper central
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, 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, C8⋊C4, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, C32⋊4C8, C6×Dic3, C6×C12, C42.S3, C24⋊C4, C6×C3⋊C8, Dic3×C12, C2×C32⋊4C8, C3⋊C8⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), C4×S3, C2×Dic3, C8⋊C4, S32, C8⋊S3, C4.Dic3, C4×Dic3, S3×Dic3, C6.D6, C42.S3, C24⋊C4, D6.Dic3, Dic32, C3⋊C8⋊Dic3
►On 96 points
Generators in S96
(1 54 23)(2 24 55)(3 56 17)(4 18 49)(5 50 19)(6 20 51)(7 52 21)(8 22 53)(9 93 69)(10 70 94)(11 95 71)(12 72 96)(13 89 65)(14 66 90)(15 91 67)(16 68 92)(25 86 35)(26 36 87)(27 88 37)(28 38 81)(29 82 39)(30 40 83)(31 84 33)(32 34 85)(41 74 59)(42 60 75)(43 76 61)(44 62 77)(45 78 63)(46 64 79)(47 80 57)(48 58 73)
(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 23 46 54 79)(2 57 24 47 55 80)(3 58 17 48 56 73)(4 59 18 41 49 74)(5 60 19 42 50 75)(6 61 20 43 51 76)(7 62 21 44 52 77)(8 63 22 45 53 78)(9 31 93 84 69 33)(10 32 94 85 70 34)(11 25 95 86 71 35)(12 26 96 87 72 36)(13 27 89 88 65 37)(14 28 90 81 66 38)(15 29 91 82 67 39)(16 30 92 83 68 40)
(1 86 46 11)(2 83 47 16)(3 88 48 13)(4 85 41 10)(5 82 42 15)(6 87 43 12)(7 84 44 9)(8 81 45 14)(17 27 73 65)(18 32 74 70)(19 29 75 67)(20 26 76 72)(21 31 77 69)(22 28 78 66)(23 25 79 71)(24 30 80 68)(33 62 93 52)(34 59 94 49)(35 64 95 54)(36 61 96 51)(37 58 89 56)(38 63 90 53)(39 60 91 50)(40 57 92 55)
G:=sub<Sym(96)| (1,54,23)(2,24,55)(3,56,17)(4,18,49)(5,50,19)(6,20,51)(7,52,21)(8,22,53)(9,93,69)(10,70,94)(11,95,71)(12,72,96)(13,89,65)(14,66,90)(15,91,67)(16,68,92)(25,86,35)(26,36,87)(27,88,37)(28,38,81)(29,82,39)(30,40,83)(31,84,33)(32,34,85)(41,74,59)(42,60,75)(43,76,61)(44,62,77)(45,78,63)(46,64,79)(47,80,57)(48,58,73), (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), (1,64,23,46,54,79)(2,57,24,47,55,80)(3,58,17,48,56,73)(4,59,18,41,49,74)(5,60,19,42,50,75)(6,61,20,43,51,76)(7,62,21,44,52,77)(8,63,22,45,53,78)(9,31,93,84,69,33)(10,32,94,85,70,34)(11,25,95,86,71,35)(12,26,96,87,72,36)(13,27,89,88,65,37)(14,28,90,81,66,38)(15,29,91,82,67,39)(16,30,92,83,68,40), (1,86,46,11)(2,83,47,16)(3,88,48,13)(4,85,41,10)(5,82,42,15)(6,87,43,12)(7,84,44,9)(8,81,45,14)(17,27,73,65)(18,32,74,70)(19,29,75,67)(20,26,76,72)(21,31,77,69)(22,28,78,66)(23,25,79,71)(24,30,80,68)(33,62,93,52)(34,59,94,49)(35,64,95,54)(36,61,96,51)(37,58,89,56)(38,63,90,53)(39,60,91,50)(40,57,92,55)>;
G:=Group( (1,54,23)(2,24,55)(3,56,17)(4,18,49)(5,50,19)(6,20,51)(7,52,21)(8,22,53)(9,93,69)(10,70,94)(11,95,71)(12,72,96)(13,89,65)(14,66,90)(15,91,67)(16,68,92)(25,86,35)(26,36,87)(27,88,37)(28,38,81)(29,82,39)(30,40,83)(31,84,33)(32,34,85)(41,74,59)(42,60,75)(43,76,61)(44,62,77)(45,78,63)(46,64,79)(47,80,57)(48,58,73), (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), (1,64,23,46,54,79)(2,57,24,47,55,80)(3,58,17,48,56,73)(4,59,18,41,49,74)(5,60,19,42,50,75)(6,61,20,43,51,76)(7,62,21,44,52,77)(8,63,22,45,53,78)(9,31,93,84,69,33)(10,32,94,85,70,34)(11,25,95,86,71,35)(12,26,96,87,72,36)(13,27,89,88,65,37)(14,28,90,81,66,38)(15,29,91,82,67,39)(16,30,92,83,68,40), (1,86,46,11)(2,83,47,16)(3,88,48,13)(4,85,41,10)(5,82,42,15)(6,87,43,12)(7,84,44,9)(8,81,45,14)(17,27,73,65)(18,32,74,70)(19,29,75,67)(20,26,76,72)(21,31,77,69)(22,28,78,66)(23,25,79,71)(24,30,80,68)(33,62,93,52)(34,59,94,49)(35,64,95,54)(36,61,96,51)(37,58,89,56)(38,63,90,53)(39,60,91,50)(40,57,92,55) );
G=PermutationGroup([[(1,54,23),(2,24,55),(3,56,17),(4,18,49),(5,50,19),(6,20,51),(7,52,21),(8,22,53),(9,93,69),(10,70,94),(11,95,71),(12,72,96),(13,89,65),(14,66,90),(15,91,67),(16,68,92),(25,86,35),(26,36,87),(27,88,37),(28,38,81),(29,82,39),(30,40,83),(31,84,33),(32,34,85),(41,74,59),(42,60,75),(43,76,61),(44,62,77),(45,78,63),(46,64,79),(47,80,57),(48,58,73)], [(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)], [(1,64,23,46,54,79),(2,57,24,47,55,80),(3,58,17,48,56,73),(4,59,18,41,49,74),(5,60,19,42,50,75),(6,61,20,43,51,76),(7,62,21,44,52,77),(8,63,22,45,53,78),(9,31,93,84,69,33),(10,32,94,85,70,34),(11,25,95,86,71,35),(12,26,96,87,72,36),(13,27,89,88,65,37),(14,28,90,81,66,38),(15,29,91,82,67,39),(16,30,92,83,68,40)], [(1,86,46,11),(2,83,47,16),(3,88,48,13),(4,85,41,10),(5,82,42,15),(6,87,43,12),(7,84,44,9),(8,81,45,14),(17,27,73,65),(18,32,74,70),(19,29,75,67),(20,26,76,72),(21,31,77,69),(22,28,78,66),(23,25,79,71),(24,30,80,68),(33,62,93,52),(34,59,94,49),(35,64,95,54),(36,61,96,51),(37,58,89,56),(38,63,90,53),(39,60,91,50),(40,57,92,55)]])
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