direct product, metabelian, supersoluble, monomial
Aliases: C3×D6⋊C8, D6⋊C24, C12.94D12, (S3×C6)⋊2C8, (C2×C24)⋊1C6, (C6×C24)⋊1C2, (C2×C24)⋊1S3, C6.26(S3×C8), C2.5(S3×C24), C6.5(C2×C24), C4.19(C3×D12), C12.61(C3×D4), C6.47(D6⋊C4), (C2×C12).456D6, (C3×C12).163D4, C32⋊7(C22⋊C8), C62.70(C2×C4), (C6×Dic3).7C4, C6.16(C8⋊S3), C6.3(C3×M4(2)), (C22×S3).2C12, C22.11(S3×C12), (C2×Dic3).4C12, (C3×C6).14M4(2), C12.144(C3⋊D4), (C6×C12).334C22, (C6×C3⋊C8)⋊6C2, (C2×C3⋊C8)⋊9C6, (C2×C8)⋊1(C3×S3), (S3×C2×C4).6C6, (S3×C2×C6).4C4, C3⋊1(C3×C22⋊C8), C2.1(C3×D6⋊C4), C2.3(C3×C8⋊S3), (S3×C2×C12).17C2, (C2×C6).81(C4×S3), (C2×C4).93(S3×C6), (C3×C6).31(C2×C8), C4.27(C3×C3⋊D4), C6.6(C3×C22⋊C4), (C2×C6).15(C2×C12), (C2×C12).123(C2×C6), (C3×C6).46(C22⋊C4), SmallGroup(288,254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6⋊C8
G = < a,b,c,d | a3=b6=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 250 in 111 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, C22×C4, C3×S3, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C2×C3⋊C8, C2×C24, C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C3×C24, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, D6⋊C8, C3×C22⋊C8, C6×C3⋊C8, C6×C24, S3×C2×C12, C3×D6⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C2×C8, M4(2), C3×S3, C24, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22⋊C8, S3×C6, S3×C8, C8⋊S3, D6⋊C4, C3×C22⋊C4, C2×C24, C3×M4(2), S3×C12, C3×D12, C3×C3⋊D4, D6⋊C8, C3×C22⋊C8, S3×C24, C3×C8⋊S3, C3×D6⋊C4, C3×D6⋊C8
►On 96 points
(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 53 62 23 43 77)(2 54 63 24 44 78)(3 55 64 17 45 79)(4 56 57 18 46 80)(5 49 58 19 47 73)(6 50 59 20 48 74)(7 51 60 21 41 75)(8 52 61 22 42 76)(9 31 96 66 86 33)(10 32 89 67 87 34)(11 25 90 68 88 35)(12 26 91 69 81 36)(13 27 92 70 82 37)(14 28 93 71 83 38)(15 29 94 72 84 39)(16 30 95 65 85 40)
(1 90)(2 36)(3 92)(4 38)(5 94)(6 40)(7 96)(8 34)(9 60)(10 76)(11 62)(12 78)(13 64)(14 80)(15 58)(16 74)(17 37)(18 93)(19 39)(20 95)(21 33)(22 89)(23 35)(24 91)(25 53)(26 44)(27 55)(28 46)(29 49)(30 48)(31 51)(32 42)(41 86)(43 88)(45 82)(47 84)(50 85)(52 87)(54 81)(56 83)(57 71)(59 65)(61 67)(63 69)(66 75)(68 77)(70 79)(72 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)
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), (1,53,62,23,43,77)(2,54,63,24,44,78)(3,55,64,17,45,79)(4,56,57,18,46,80)(5,49,58,19,47,73)(6,50,59,20,48,74)(7,51,60,21,41,75)(8,52,61,22,42,76)(9,31,96,66,86,33)(10,32,89,67,87,34)(11,25,90,68,88,35)(12,26,91,69,81,36)(13,27,92,70,82,37)(14,28,93,71,83,38)(15,29,94,72,84,39)(16,30,95,65,85,40), (1,90)(2,36)(3,92)(4,38)(5,94)(6,40)(7,96)(8,34)(9,60)(10,76)(11,62)(12,78)(13,64)(14,80)(15,58)(16,74)(17,37)(18,93)(19,39)(20,95)(21,33)(22,89)(23,35)(24,91)(25,53)(26,44)(27,55)(28,46)(29,49)(30,48)(31,51)(32,42)(41,86)(43,88)(45,82)(47,84)(50,85)(52,87)(54,81)(56,83)(57,71)(59,65)(61,67)(63,69)(66,75)(68,77)(70,79)(72,73), (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)>;
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), (1,53,62,23,43,77)(2,54,63,24,44,78)(3,55,64,17,45,79)(4,56,57,18,46,80)(5,49,58,19,47,73)(6,50,59,20,48,74)(7,51,60,21,41,75)(8,52,61,22,42,76)(9,31,96,66,86,33)(10,32,89,67,87,34)(11,25,90,68,88,35)(12,26,91,69,81,36)(13,27,92,70,82,37)(14,28,93,71,83,38)(15,29,94,72,84,39)(16,30,95,65,85,40), (1,90)(2,36)(3,92)(4,38)(5,94)(6,40)(7,96)(8,34)(9,60)(10,76)(11,62)(12,78)(13,64)(14,80)(15,58)(16,74)(17,37)(18,93)(19,39)(20,95)(21,33)(22,89)(23,35)(24,91)(25,53)(26,44)(27,55)(28,46)(29,49)(30,48)(31,51)(32,42)(41,86)(43,88)(45,82)(47,84)(50,85)(52,87)(54,81)(56,83)(57,71)(59,65)(61,67)(63,69)(66,75)(68,77)(70,79)(72,73), (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) );
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)], [(1,53,62,23,43,77),(2,54,63,24,44,78),(3,55,64,17,45,79),(4,56,57,18,46,80),(5,49,58,19,47,73),(6,50,59,20,48,74),(7,51,60,21,41,75),(8,52,61,22,42,76),(9,31,96,66,86,33),(10,32,89,67,87,34),(11,25,90,68,88,35),(12,26,91,69,81,36),(13,27,92,70,82,37),(14,28,93,71,83,38),(15,29,94,72,84,39),(16,30,95,65,85,40)], [(1,90),(2,36),(3,92),(4,38),(5,94),(6,40),(7,96),(8,34),(9,60),(10,76),(11,62),(12,78),(13,64),(14,80),(15,58),(16,74),(17,37),(18,93),(19,39),(20,95),(21,33),(22,89),(23,35),(24,91),(25,53),(26,44),(27,55),(28,46),(29,49),(30,48),(31,51),(32,42),(41,86),(43,88),(45,82),(47,84),(50,85),(52,87),(54,81),(56,83),(57,71),(59,65),(61,67),(63,69),(66,75),(68,77),(70,79),(72,73)], [(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)]])
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 | 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 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D4 | D6 | M4(2) | C3×S3 | D12 | C3⋊D4 | C3×D4 | C4×S3 | S3×C6 | S3×C8 | C8⋊S3 | C3×M4(2) | C3×D12 | C3×C3⋊D4 | S3×C12 | S3×C24 | C3×C8⋊S3 |
| kernel | C3×D6⋊C8 | C6×C3⋊C8 | C6×C24 | S3×C2×C12 | D6⋊C8 | C6×Dic3 | S3×C2×C6 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | S3×C6 | C2×Dic3 | C22×S3 | D6 | C2×C24 | C3×C12 | C2×C12 | C3×C6 | C2×C8 | C12 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×D6⋊C8 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 65 |
| 72 | 0 | 0 |
| 0 | 0 | 65 |
| 0 | 9 | 0 |
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,9,0,0,0,65],[72,0,0,0,0,9,0,65,0],[10,0,0,0,1,0,0,0,72] >;
C3×D6⋊C8 in GAP, Magma, Sage, TeX
C_3\times D_6\rtimes C_8
% in TeX
G:=Group("C3xD6:C8"); // GroupNames label
G:=SmallGroup(288,254);
// by ID
G=gap.SmallGroup(288,254);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,136,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations