direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×C3⋊D4, C24⋊32D4, C3⋊5(C8×D4), D6⋊4(C2×C8), D6⋊C8⋊42C2, (C22×C8)⋊6S3, C22⋊3(S3×C8), C6.77(C4×D4), D6⋊C4.20C4, Dic3⋊2(C2×C8), (C2×C8).345D6, Dic3⋊C8⋊43C2, (C22×C24)⋊18C2, (C8×Dic3)⋊26C2, C2.5(C8○D12), C6.17(C8○D4), C12.435(C2×D4), C23.38(C4×S3), C6.20(C22×C8), Dic3⋊C4.20C4, (C22×C4).417D6, C12.251(C4○D4), C4.135(C4○D12), C12.55D4⋊32C2, (C2×C24).353C22, (C2×C12).859C23, C6.D4.15C4, (C22×C12).560C22, (C4×Dic3).282C22, (S3×C2×C8)⋊24C2, (C2×C6)⋊5(C2×C8), C2.20(S3×C2×C8), C2.3(C4×C3⋊D4), (C2×C4).93(C4×S3), C22.61(S3×C2×C4), (C4×C3⋊D4).19C2, (C2×C3⋊D4).16C4, C4.125(C2×C3⋊D4), (C2×C12).209(C2×C4), (C2×C3⋊C8).320C22, (S3×C2×C4).283C22, (C22×C6).94(C2×C4), (C22×S3).43(C2×C4), (C2×C4).801(C22×S3), (C2×C6).129(C22×C4), (C2×Dic3).66(C2×C4), SmallGroup(192,668)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×C3⋊D4
G = < a,b,c,d | a8=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 280 in 134 conjugacy classes, 63 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8×D4, C8×Dic3, Dic3⋊C8, D6⋊C8, C12.55D4, S3×C2×C8, C4×C3⋊D4, C22×C24, C8×C3⋊D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C2×C8, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C22×C8, C8○D4, S3×C8, S3×C2×C4, C4○D12, C2×C3⋊D4, C8×D4, S3×C2×C8, C8○D12, C4×C3⋊D4, C8×C3⋊D4
►On 96 points
(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 21 43)(2 22 44)(3 23 45)(4 24 46)(5 17 47)(6 18 48)(7 19 41)(8 20 42)(9 39 65)(10 40 66)(11 33 67)(12 34 68)(13 35 69)(14 36 70)(15 37 71)(16 38 72)(25 64 89)(26 57 90)(27 58 91)(28 59 92)(29 60 93)(30 61 94)(31 62 95)(32 63 96)(49 79 85)(50 80 86)(51 73 87)(52 74 88)(53 75 81)(54 76 82)(55 77 83)(56 78 84)
(1 31 55 69)(2 32 56 70)(3 25 49 71)(4 26 50 72)(5 27 51 65)(6 28 52 66)(7 29 53 67)(8 30 54 68)(9 47 58 87)(10 48 59 88)(11 41 60 81)(12 42 61 82)(13 43 62 83)(14 44 63 84)(15 45 64 85)(16 46 57 86)(17 91 73 39)(18 92 74 40)(19 93 75 33)(20 94 76 34)(21 95 77 35)(22 96 78 36)(23 89 79 37)(24 90 80 38)
(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 89)(16 90)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 71)(26 72)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(73 87)(74 88)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)
G:=sub<Sym(96)| (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), (1,21,43)(2,22,44)(3,23,45)(4,24,46)(5,17,47)(6,18,48)(7,19,41)(8,20,42)(9,39,65)(10,40,66)(11,33,67)(12,34,68)(13,35,69)(14,36,70)(15,37,71)(16,38,72)(25,64,89)(26,57,90)(27,58,91)(28,59,92)(29,60,93)(30,61,94)(31,62,95)(32,63,96)(49,79,85)(50,80,86)(51,73,87)(52,74,88)(53,75,81)(54,76,82)(55,77,83)(56,78,84), (1,31,55,69)(2,32,56,70)(3,25,49,71)(4,26,50,72)(5,27,51,65)(6,28,52,66)(7,29,53,67)(8,30,54,68)(9,47,58,87)(10,48,59,88)(11,41,60,81)(12,42,61,82)(13,43,62,83)(14,44,63,84)(15,45,64,85)(16,46,57,86)(17,91,73,39)(18,92,74,40)(19,93,75,33)(20,94,76,34)(21,95,77,35)(22,96,78,36)(23,89,79,37)(24,90,80,38), (9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,71)(26,72)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(73,87)(74,88)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)>;
G:=Group( (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), (1,21,43)(2,22,44)(3,23,45)(4,24,46)(5,17,47)(6,18,48)(7,19,41)(8,20,42)(9,39,65)(10,40,66)(11,33,67)(12,34,68)(13,35,69)(14,36,70)(15,37,71)(16,38,72)(25,64,89)(26,57,90)(27,58,91)(28,59,92)(29,60,93)(30,61,94)(31,62,95)(32,63,96)(49,79,85)(50,80,86)(51,73,87)(52,74,88)(53,75,81)(54,76,82)(55,77,83)(56,78,84), (1,31,55,69)(2,32,56,70)(3,25,49,71)(4,26,50,72)(5,27,51,65)(6,28,52,66)(7,29,53,67)(8,30,54,68)(9,47,58,87)(10,48,59,88)(11,41,60,81)(12,42,61,82)(13,43,62,83)(14,44,63,84)(15,45,64,85)(16,46,57,86)(17,91,73,39)(18,92,74,40)(19,93,75,33)(20,94,76,34)(21,95,77,35)(22,96,78,36)(23,89,79,37)(24,90,80,38), (9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,71)(26,72)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(73,87)(74,88)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86) );
G=PermutationGroup([[(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)], [(1,21,43),(2,22,44),(3,23,45),(4,24,46),(5,17,47),(6,18,48),(7,19,41),(8,20,42),(9,39,65),(10,40,66),(11,33,67),(12,34,68),(13,35,69),(14,36,70),(15,37,71),(16,38,72),(25,64,89),(26,57,90),(27,58,91),(28,59,92),(29,60,93),(30,61,94),(31,62,95),(32,63,96),(49,79,85),(50,80,86),(51,73,87),(52,74,88),(53,75,81),(54,76,82),(55,77,83),(56,78,84)], [(1,31,55,69),(2,32,56,70),(3,25,49,71),(4,26,50,72),(5,27,51,65),(6,28,52,66),(7,29,53,67),(8,30,54,68),(9,47,58,87),(10,48,59,88),(11,41,60,81),(12,42,61,82),(13,43,62,83),(14,44,63,84),(15,45,64,85),(16,46,57,86),(17,91,73,39),(18,92,74,40),(19,93,75,33),(20,94,76,34),(21,95,77,35),(22,96,78,36),(23,89,79,37),(24,90,80,38)], [(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,89),(16,90),(17,47),(18,48),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,71),(26,72),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(73,87),(74,88),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4×S3 | C8○D4 | C4○D12 | S3×C8 | C8○D12 |
| kernel | C8×C3⋊D4 | C8×Dic3 | Dic3⋊C8 | D6⋊C8 | C12.55D4 | S3×C2×C8 | C4×C3⋊D4 | C22×C24 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C3⋊D4 | C22×C8 | C24 | C2×C8 | C22×C4 | C12 | C8 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C8×C3⋊D4 ►in GL3(𝔽73) generated by
| 22 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 72 | 72 |
| 0 | 1 | 0 |
| 72 | 0 | 0 |
| 0 | 43 | 13 |
| 0 | 43 | 30 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 72 | 72 |
G:=sub<GL(3,GF(73))| [22,0,0,0,1,0,0,0,1],[1,0,0,0,72,1,0,72,0],[72,0,0,0,43,43,0,13,30],[72,0,0,0,1,72,0,0,72] >;
C8×C3⋊D4 in GAP, Magma, Sage, TeX
C_8\times C_3\rtimes D_4
% in TeX
G:=Group("C8xC3:D4"); // GroupNames label
G:=SmallGroup(192,668);
// by ID
G=gap.SmallGroup(192,668);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations