direct product, metabelian, supersoluble, monomial
Aliases: C6×C8⋊S3, C24⋊29D6, C8⋊9(S3×C6), C24⋊12(C2×C6), (C2×C24)⋊14S3, (C6×C24)⋊19C2, (C2×C24)⋊11C6, (S3×C12).8C4, (C4×S3).3C12, C4.24(S3×C12), D6.5(C2×C12), C3⋊1(C6×M4(2)), (C3×C6)⋊5M4(2), C6⋊1(C3×M4(2)), C12.115(C4×S3), C12.29(C2×C12), (C3×C24)⋊32C22, (C2×C12).461D6, C62.77(C2×C4), (C22×S3).3C12, C12.36(C22×C6), C6.13(C22×C12), C22.14(S3×C12), Dic3.6(C2×C12), (C2×Dic3).5C12, (C6×Dic3).13C4, C32⋊10(C2×M4(2)), (S3×C12).58C22, (C6×C12).348C22, (C3×C12).168C23, C12.224(C22×S3), (C6×C3⋊C8)⋊23C2, (C2×C3⋊C8)⋊11C6, C3⋊C8⋊10(C2×C6), (C2×C8)⋊6(C3×S3), (S3×C2×C6).8C4, C4.36(S3×C2×C6), (S3×C2×C4).10C6, C2.14(S3×C2×C12), C6.112(S3×C2×C4), (C3×C3⋊C8)⋊41C22, (S3×C2×C12).23C2, (C2×C6).84(C4×S3), (C2×C4).99(S3×C6), (S3×C6).23(C2×C4), (C4×S3).14(C2×C6), (C2×C6).18(C2×C12), (C3×C12).114(C2×C4), (C2×C12).129(C2×C6), (C3×C6).84(C22×C4), (C3×Dic3).28(C2×C4), SmallGroup(288,671)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C8⋊S3
G = < a,b,c,d | a6=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 282 in 147 conjugacy classes, 82 normal (38 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×M4(2), C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C2×C3⋊C8, C2×C24, C2×C24, C3×M4(2), S3×C2×C4, C22×C12, C3×C3⋊C8, C3×C24, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C2×C8⋊S3, C6×M4(2), C3×C8⋊S3, C6×C3⋊C8, C6×C24, S3×C2×C12, C6×C8⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, M4(2), C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C2×M4(2), S3×C6, C8⋊S3, C3×M4(2), S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, C2×C8⋊S3, C6×M4(2), C3×C8⋊S3, S3×C2×C12, C6×C8⋊S3
►On 96 points
(1 64 25 56 35 42)(2 57 26 49 36 43)(3 58 27 50 37 44)(4 59 28 51 38 45)(5 60 29 52 39 46)(6 61 30 53 40 47)(7 62 31 54 33 48)(8 63 32 55 34 41)(9 82 96 74 23 72)(10 83 89 75 24 65)(11 84 90 76 17 66)(12 85 91 77 18 67)(13 86 92 78 19 68)(14 87 93 79 20 69)(15 88 94 80 21 70)(16 81 95 73 22 71)
(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 35 25)(2 36 26)(3 37 27)(4 38 28)(5 39 29)(6 40 30)(7 33 31)(8 34 32)(9 96 23)(10 89 24)(11 90 17)(12 91 18)(13 92 19)(14 93 20)(15 94 21)(16 95 22)(41 55 63)(42 56 64)(43 49 57)(44 50 58)(45 51 59)(46 52 60)(47 53 61)(48 54 62)(65 83 75)(66 84 76)(67 85 77)(68 86 78)(69 87 79)(70 88 80)(71 81 73)(72 82 74)
(1 76)(2 73)(3 78)(4 75)(5 80)(6 77)(7 74)(8 79)(9 54)(10 51)(11 56)(12 53)(13 50)(14 55)(15 52)(16 49)(17 64)(18 61)(19 58)(20 63)(21 60)(22 57)(23 62)(24 59)(25 66)(26 71)(27 68)(28 65)(29 70)(30 67)(31 72)(32 69)(33 82)(34 87)(35 84)(36 81)(37 86)(38 83)(39 88)(40 85)(41 93)(42 90)(43 95)(44 92)(45 89)(46 94)(47 91)(48 96)
G:=sub<Sym(96)| (1,64,25,56,35,42)(2,57,26,49,36,43)(3,58,27,50,37,44)(4,59,28,51,38,45)(5,60,29,52,39,46)(6,61,30,53,40,47)(7,62,31,54,33,48)(8,63,32,55,34,41)(9,82,96,74,23,72)(10,83,89,75,24,65)(11,84,90,76,17,66)(12,85,91,77,18,67)(13,86,92,78,19,68)(14,87,93,79,20,69)(15,88,94,80,21,70)(16,81,95,73,22,71), (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), (1,35,25)(2,36,26)(3,37,27)(4,38,28)(5,39,29)(6,40,30)(7,33,31)(8,34,32)(9,96,23)(10,89,24)(11,90,17)(12,91,18)(13,92,19)(14,93,20)(15,94,21)(16,95,22)(41,55,63)(42,56,64)(43,49,57)(44,50,58)(45,51,59)(46,52,60)(47,53,61)(48,54,62)(65,83,75)(66,84,76)(67,85,77)(68,86,78)(69,87,79)(70,88,80)(71,81,73)(72,82,74), (1,76)(2,73)(3,78)(4,75)(5,80)(6,77)(7,74)(8,79)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,64)(18,61)(19,58)(20,63)(21,60)(22,57)(23,62)(24,59)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(33,82)(34,87)(35,84)(36,81)(37,86)(38,83)(39,88)(40,85)(41,93)(42,90)(43,95)(44,92)(45,89)(46,94)(47,91)(48,96)>;
G:=Group( (1,64,25,56,35,42)(2,57,26,49,36,43)(3,58,27,50,37,44)(4,59,28,51,38,45)(5,60,29,52,39,46)(6,61,30,53,40,47)(7,62,31,54,33,48)(8,63,32,55,34,41)(9,82,96,74,23,72)(10,83,89,75,24,65)(11,84,90,76,17,66)(12,85,91,77,18,67)(13,86,92,78,19,68)(14,87,93,79,20,69)(15,88,94,80,21,70)(16,81,95,73,22,71), (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), (1,35,25)(2,36,26)(3,37,27)(4,38,28)(5,39,29)(6,40,30)(7,33,31)(8,34,32)(9,96,23)(10,89,24)(11,90,17)(12,91,18)(13,92,19)(14,93,20)(15,94,21)(16,95,22)(41,55,63)(42,56,64)(43,49,57)(44,50,58)(45,51,59)(46,52,60)(47,53,61)(48,54,62)(65,83,75)(66,84,76)(67,85,77)(68,86,78)(69,87,79)(70,88,80)(71,81,73)(72,82,74), (1,76)(2,73)(3,78)(4,75)(5,80)(6,77)(7,74)(8,79)(9,54)(10,51)(11,56)(12,53)(13,50)(14,55)(15,52)(16,49)(17,64)(18,61)(19,58)(20,63)(21,60)(22,57)(23,62)(24,59)(25,66)(26,71)(27,68)(28,65)(29,70)(30,67)(31,72)(32,69)(33,82)(34,87)(35,84)(36,81)(37,86)(38,83)(39,88)(40,85)(41,93)(42,90)(43,95)(44,92)(45,89)(46,94)(47,91)(48,96) );
G=PermutationGroup([[(1,64,25,56,35,42),(2,57,26,49,36,43),(3,58,27,50,37,44),(4,59,28,51,38,45),(5,60,29,52,39,46),(6,61,30,53,40,47),(7,62,31,54,33,48),(8,63,32,55,34,41),(9,82,96,74,23,72),(10,83,89,75,24,65),(11,84,90,76,17,66),(12,85,91,77,18,67),(13,86,92,78,19,68),(14,87,93,79,20,69),(15,88,94,80,21,70),(16,81,95,73,22,71)], [(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)], [(1,35,25),(2,36,26),(3,37,27),(4,38,28),(5,39,29),(6,40,30),(7,33,31),(8,34,32),(9,96,23),(10,89,24),(11,90,17),(12,91,18),(13,92,19),(14,93,20),(15,94,21),(16,95,22),(41,55,63),(42,56,64),(43,49,57),(44,50,58),(45,51,59),(46,52,60),(47,53,61),(48,54,62),(65,83,75),(66,84,76),(67,85,77),(68,86,78),(69,87,79),(70,88,80),(71,81,73),(72,82,74)], [(1,76),(2,73),(3,78),(4,75),(5,80),(6,77),(7,74),(8,79),(9,54),(10,51),(11,56),(12,53),(13,50),(14,55),(15,52),(16,49),(17,64),(18,61),(19,58),(20,63),(21,60),(22,57),(23,62),(24,59),(25,66),(26,71),(27,68),(28,65),(29,70),(30,67),(31,72),(32,69),(33,82),(34,87),(35,84),(36,81),(37,86),(38,83),(39,88),(40,85),(41,93),(42,90),(43,95),(44,92),(45,89),(46,94),(47,91),(48,96)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 12I | ··· | 12T | 12U | 12V | 12W | 12X | 24A | ··· | 24AF | 24AG | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | S3 | D6 | D6 | M4(2) | C3×S3 | C4×S3 | C4×S3 | S3×C6 | S3×C6 | C8⋊S3 | C3×M4(2) | S3×C12 | S3×C12 | C3×C8⋊S3 |
| kernel | C6×C8⋊S3 | C3×C8⋊S3 | C6×C3⋊C8 | C6×C24 | S3×C2×C12 | C2×C8⋊S3 | S3×C12 | C6×Dic3 | S3×C2×C6 | C8⋊S3 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | C2×C24 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 8 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C6×C8⋊S3 ►in GL4(𝔽73) generated by
| 65 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 63 | 17 |
| 0 | 0 | 0 | 10 |
| 8 | 64 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 33 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 63 | 1 | 0 | 0 |
| 0 | 0 | 28 | 71 |
| 0 | 0 | 63 | 45 |
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,8,0,0,0,0,8],[46,0,0,0,0,46,0,0,0,0,63,0,0,0,17,10],[8,0,0,0,64,64,0,0,0,0,8,0,0,0,33,64],[72,63,0,0,0,1,0,0,0,0,28,63,0,0,71,45] >;
C6×C8⋊S3 in GAP, Magma, Sage, TeX
C_6\times C_8\rtimes S_3
% in TeX
G:=Group("C6xC8:S3"); // GroupNames label
G:=SmallGroup(288,671);
// by ID
G=gap.SmallGroup(288,671);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,142,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations