direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3⋊C8, Dic3⋊C24, C12.35Dic6, C6.25(S3×C8), C2.4(S3×C24), (C6×C24).1C2, (C2×C24).5S3, C6.4(C2×C24), (C2×C24).5C6, C12.8(C3×Q8), C32⋊10(C4⋊C8), (C3×Dic3)⋊3C8, C12.60(C3×D4), (C3×C12).26Q8, C4.8(C3×Dic6), (C3×C12).162D4, (C2×C12).454D6, C62.68(C2×C4), C22.9(S3×C12), (C4×Dic3).5C6, C6.14(C8⋊S3), C6.1(C3×M4(2)), (C6×Dic3).12C4, (C2×Dic3).2C12, (C3×C6).12M4(2), C12.143(C3⋊D4), C6.23(Dic3⋊C4), (C6×C12).332C22, (Dic3×C12).17C2, C3⋊2(C3×C4⋊C8), C6.4(C3×C4⋊C4), (C6×C3⋊C8).7C2, (C2×C3⋊C8).9C6, (C2×C8).1(C3×S3), C2.1(C3×C8⋊S3), (C2×C4).91(S3×C6), (C3×C6).30(C2×C8), (C2×C6).79(C4×S3), C4.26(C3×C3⋊D4), (C3×C6).30(C4⋊C4), (C2×C6).13(C2×C12), C2.1(C3×Dic3⋊C4), (C2×C12).121(C2×C6), SmallGroup(288,248)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4⋊C8, C3×Dic3, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, C2×C24, C3×C3⋊C8, C3×C24, C6×Dic3, C6×C12, Dic3⋊C8, C3×C4⋊C8, C6×C3⋊C8, Dic3×C12, C6×C24, C3×Dic3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C2×C8, M4(2), C3×S3, C24, 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
►On 96 points
(1 23 43)(2 24 44)(3 17 45)(4 18 46)(5 19 47)(6 20 48)(7 21 41)(8 22 42)(9 57 96)(10 58 89)(11 59 90)(12 60 91)(13 61 92)(14 62 93)(15 63 94)(16 64 95)(25 35 68)(26 36 69)(27 37 70)(28 38 71)(29 39 72)(30 40 65)(31 33 66)(32 34 67)(49 73 87)(50 74 88)(51 75 81)(52 76 82)(53 77 83)(54 78 84)(55 79 85)(56 80 86)
(1 77 43 53 23 83)(2 78 44 54 24 84)(3 79 45 55 17 85)(4 80 46 56 18 86)(5 73 47 49 19 87)(6 74 48 50 20 88)(7 75 41 51 21 81)(8 76 42 52 22 82)(9 33 57 66 96 31)(10 34 58 67 89 32)(11 35 59 68 90 25)(12 36 60 69 91 26)(13 37 61 70 92 27)(14 38 62 71 93 28)(15 39 63 72 94 29)(16 40 64 65 95 30)
(1 11 53 68)(2 69 54 12)(3 13 55 70)(4 71 56 14)(5 15 49 72)(6 65 50 16)(7 9 51 66)(8 67 52 10)(17 61 79 27)(18 28 80 62)(19 63 73 29)(20 30 74 64)(21 57 75 31)(22 32 76 58)(23 59 77 25)(24 26 78 60)(33 41 96 81)(34 82 89 42)(35 43 90 83)(36 84 91 44)(37 45 92 85)(38 86 93 46)(39 47 94 87)(40 88 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)| (1,23,43)(2,24,44)(3,17,45)(4,18,46)(5,19,47)(6,20,48)(7,21,41)(8,22,42)(9,57,96)(10,58,89)(11,59,90)(12,60,91)(13,61,92)(14,62,93)(15,63,94)(16,64,95)(25,35,68)(26,36,69)(27,37,70)(28,38,71)(29,39,72)(30,40,65)(31,33,66)(32,34,67)(49,73,87)(50,74,88)(51,75,81)(52,76,82)(53,77,83)(54,78,84)(55,79,85)(56,80,86), (1,77,43,53,23,83)(2,78,44,54,24,84)(3,79,45,55,17,85)(4,80,46,56,18,86)(5,73,47,49,19,87)(6,74,48,50,20,88)(7,75,41,51,21,81)(8,76,42,52,22,82)(9,33,57,66,96,31)(10,34,58,67,89,32)(11,35,59,68,90,25)(12,36,60,69,91,26)(13,37,61,70,92,27)(14,38,62,71,93,28)(15,39,63,72,94,29)(16,40,64,65,95,30), (1,11,53,68)(2,69,54,12)(3,13,55,70)(4,71,56,14)(5,15,49,72)(6,65,50,16)(7,9,51,66)(8,67,52,10)(17,61,79,27)(18,28,80,62)(19,63,73,29)(20,30,74,64)(21,57,75,31)(22,32,76,58)(23,59,77,25)(24,26,78,60)(33,41,96,81)(34,82,89,42)(35,43,90,83)(36,84,91,44)(37,45,92,85)(38,86,93,46)(39,47,94,87)(40,88,95,48), (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( (1,23,43)(2,24,44)(3,17,45)(4,18,46)(5,19,47)(6,20,48)(7,21,41)(8,22,42)(9,57,96)(10,58,89)(11,59,90)(12,60,91)(13,61,92)(14,62,93)(15,63,94)(16,64,95)(25,35,68)(26,36,69)(27,37,70)(28,38,71)(29,39,72)(30,40,65)(31,33,66)(32,34,67)(49,73,87)(50,74,88)(51,75,81)(52,76,82)(53,77,83)(54,78,84)(55,79,85)(56,80,86), (1,77,43,53,23,83)(2,78,44,54,24,84)(3,79,45,55,17,85)(4,80,46,56,18,86)(5,73,47,49,19,87)(6,74,48,50,20,88)(7,75,41,51,21,81)(8,76,42,52,22,82)(9,33,57,66,96,31)(10,34,58,67,89,32)(11,35,59,68,90,25)(12,36,60,69,91,26)(13,37,61,70,92,27)(14,38,62,71,93,28)(15,39,63,72,94,29)(16,40,64,65,95,30), (1,11,53,68)(2,69,54,12)(3,13,55,70)(4,71,56,14)(5,15,49,72)(6,65,50,16)(7,9,51,66)(8,67,52,10)(17,61,79,27)(18,28,80,62)(19,63,73,29)(20,30,74,64)(21,57,75,31)(22,32,76,58)(23,59,77,25)(24,26,78,60)(33,41,96,81)(34,82,89,42)(35,43,90,83)(36,84,91,44)(37,45,92,85)(38,86,93,46)(39,47,94,87)(40,88,95,48), (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([[(1,23,43),(2,24,44),(3,17,45),(4,18,46),(5,19,47),(6,20,48),(7,21,41),(8,22,42),(9,57,96),(10,58,89),(11,59,90),(12,60,91),(13,61,92),(14,62,93),(15,63,94),(16,64,95),(25,35,68),(26,36,69),(27,37,70),(28,38,71),(29,39,72),(30,40,65),(31,33,66),(32,34,67),(49,73,87),(50,74,88),(51,75,81),(52,76,82),(53,77,83),(54,78,84),(55,79,85),(56,80,86)], [(1,77,43,53,23,83),(2,78,44,54,24,84),(3,79,45,55,17,85),(4,80,46,56,18,86),(5,73,47,49,19,87),(6,74,48,50,20,88),(7,75,41,51,21,81),(8,76,42,52,22,82),(9,33,57,66,96,31),(10,34,58,67,89,32),(11,35,59,68,90,25),(12,36,60,69,91,26),(13,37,61,70,92,27),(14,38,62,71,93,28),(15,39,63,72,94,29),(16,40,64,65,95,30)], [(1,11,53,68),(2,69,54,12),(3,13,55,70),(4,71,56,14),(5,15,49,72),(6,65,50,16),(7,9,51,66),(8,67,52,10),(17,61,79,27),(18,28,80,62),(19,63,73,29),(20,30,74,64),(21,57,75,31),(22,32,76,58),(23,59,77,25),(24,26,78,60),(33,41,96,81),(34,82,89,42),(35,43,90,83),(36,84,91,44),(37,45,92,85),(38,86,93,46),(39,47,94,87),(40,88,95,48)], [(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