direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12.C4, M4(2)⋊25D6, C24.51C23, C12.69C24, C6⋊2(C8○D4), C4○D12.4C4, (C2×C8).279D6, C3⋊C8.35C23, (S3×C8)⋊22C22, D12.28(C2×C4), (C2×D12).16C4, C4.68(S3×C23), C23.35(C4×S3), C6.32(C23×C4), C8.43(C22×S3), C8⋊S3⋊18C22, (C6×M4(2))⋊14C2, (C2×M4(2))⋊17S3, (C4×S3).35C23, C12.92(C22×C4), (C2×Dic6).16C4, Dic6.29(C2×C4), D6.13(C22×C4), (C22×C4).390D6, (C2×C24).281C22, (C2×C12).882C23, C4○D12.49C22, (C3×M4(2))⋊30C22, Dic3.13(C22×C4), (C22×C12).264C22, C3⋊2(C2×C8○D4), (S3×C2×C8)⋊29C2, C4.123(S3×C2×C4), C22.8(S3×C2×C4), (C22×C3⋊C8)⋊10C2, (C2×C3⋊C8)⋊33C22, (C2×C4).87(C4×S3), C3⋊D4.3(C2×C4), (C2×C8⋊S3)⋊27C2, C2.33(S3×C22×C4), (C4×S3).24(C2×C4), (C2×C3⋊D4).14C4, (C2×C12).131(C2×C4), (C2×C4○D12).21C2, (S3×C2×C4).303C22, (C22×C6).78(C2×C4), (C2×C6).25(C22×C4), (C22×S3).46(C2×C4), (C2×C4).825(C22×S3), (C2×Dic3).72(C2×C4), SmallGroup(192,1303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D12.C4
G = < a,b,c,d | a2=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >
Subgroups: 504 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C2×C8○D4, S3×C2×C8, C2×C8⋊S3, D12.C4, C22×C3⋊C8, C6×M4(2), C2×C4○D12, C2×D12.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C8○D4, C23×C4, S3×C2×C4, S3×C23, C2×C8○D4, D12.C4, S3×C22×C4, C2×D12.C4
►On 96 points
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 82)(14 83)(15 84)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(49 68)(50 69)(51 70)(52 71)(53 72)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 76)(14 75)(15 74)(16 73)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 72)(58 71)(59 70)(60 69)
(1 54 85 15 7 60 91 21)(2 49 86 22 8 55 92 16)(3 56 87 17 9 50 93 23)(4 51 88 24 10 57 94 18)(5 58 89 19 11 52 95 13)(6 53 90 14 12 59 96 20)(25 76 37 71 31 82 43 65)(26 83 38 66 32 77 44 72)(27 78 39 61 33 84 45 67)(28 73 40 68 34 79 46 62)(29 80 41 63 35 74 47 69)(30 75 42 70 36 81 48 64)
G:=sub<Sym(96)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,82)(14,83)(15,84)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(49,68)(50,69)(51,70)(52,71)(53,72)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,48)(13,76)(14,75)(15,74)(16,73)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,89)(26,88)(27,87)(28,86)(29,85)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,72)(58,71)(59,70)(60,69), (1,54,85,15,7,60,91,21)(2,49,86,22,8,55,92,16)(3,56,87,17,9,50,93,23)(4,51,88,24,10,57,94,18)(5,58,89,19,11,52,95,13)(6,53,90,14,12,59,96,20)(25,76,37,71,31,82,43,65)(26,83,38,66,32,77,44,72)(27,78,39,61,33,84,45,67)(28,73,40,68,34,79,46,62)(29,80,41,63,35,74,47,69)(30,75,42,70,36,81,48,64)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,37)(12,38)(13,82)(14,83)(15,84)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(49,68)(50,69)(51,70)(52,71)(53,72)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,48)(13,76)(14,75)(15,74)(16,73)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,89)(26,88)(27,87)(28,86)(29,85)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,72)(58,71)(59,70)(60,69), (1,54,85,15,7,60,91,21)(2,49,86,22,8,55,92,16)(3,56,87,17,9,50,93,23)(4,51,88,24,10,57,94,18)(5,58,89,19,11,52,95,13)(6,53,90,14,12,59,96,20)(25,76,37,71,31,82,43,65)(26,83,38,66,32,77,44,72)(27,78,39,61,33,84,45,67)(28,73,40,68,34,79,46,62)(29,80,41,63,35,74,47,69)(30,75,42,70,36,81,48,64) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,37),(12,38),(13,82),(14,83),(15,84),(16,73),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,85),(34,86),(35,87),(36,88),(49,68),(50,69),(51,70),(52,71),(53,72),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,48),(13,76),(14,75),(15,74),(16,73),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,89),(26,88),(27,87),(28,86),(29,85),(30,96),(31,95),(32,94),(33,93),(34,92),(35,91),(36,90),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,72),(58,71),(59,70),(60,69)], [(1,54,85,15,7,60,91,21),(2,49,86,22,8,55,92,16),(3,56,87,17,9,50,93,23),(4,51,88,24,10,57,94,18),(5,58,89,19,11,52,95,13),(6,53,90,14,12,59,96,20),(25,76,37,71,31,82,43,65),(26,83,38,66,32,77,44,72),(27,78,39,61,33,84,45,67),(28,73,40,68,34,79,46,62),(29,80,41,63,35,74,47,69),(30,75,42,70,36,81,48,64)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | D12.C4 |
| kernel | C2×D12.C4 | S3×C2×C8 | C2×C8⋊S3 | D12.C4 | C22×C3⋊C8 | C6×M4(2) | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C2×C3⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 1 | 2 | 4 | 1 | 6 | 2 | 8 | 4 |
Matrix representation of C2×D12.C4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 55 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
| 51 | 34 | 0 | 0 |
| 66 | 22 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,55,27,0,0,0,0,1,1,0,0,72,0],[72,3,0,0,0,1,0,0,0,0,1,1,0,0,0,72],[51,66,0,0,34,22,0,0,0,0,72,0,0,0,0,72] >;
C2×D12.C4 in GAP, Magma, Sage, TeX
C_2\times D_{12}.C_4 % in TeX
G:=Group("C2xD12.C4"); // GroupNames label
G:=SmallGroup(192,1303);
// by ID
G=gap.SmallGroup(192,1303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,297,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations