metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊6D12, C4⋊2(D6⋊C4), (C2×D12)⋊7C4, (C2×C12)⋊28D4, (C2×C42)⋊7S3, C6.18(C4×D4), C2.19(C4×D12), C12⋊4(C22⋊C4), C6.56(C4⋊D4), C6.10(C4⋊1D4), C2.2(C4⋊D12), C2.2(C12⋊7D4), (C22×D12).4C2, C22.39(C2×D12), (C22×C4).415D6, C2.3(C42⋊7S3), C6.12(C4.4D4), C22.48(C4○D12), (S3×C23).11C22, C23.282(C22×S3), (C22×C6).314C23, C3⋊1(C24.3C22), (C22×C12).478C22, (C22×Dic3).32C22, (C2×C4×C12)⋊7C2, (C2×D6⋊C4)⋊2C2, C2.6(C2×D6⋊C4), (C2×C4⋊Dic3)⋊7C2, (C2×C4).111(C4×S3), (C2×C6).428(C2×D4), C6.33(C2×C22⋊C4), C22.119(S3×C2×C4), (C2×C12).225(C2×C4), (C2×C6).73(C4○D4), C22.43(C2×C3⋊D4), (C2×C4).239(C3⋊D4), (C22×S3).18(C2×C4), (C2×C6).100(C22×C4), SmallGroup(192,498)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊6D12
G = < a,b,c,d | a2=b4=c12=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=c-1 >
Subgroups: 824 in 258 conjugacy classes, 87 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4⋊Dic3, D6⋊C4, C4×C12, C2×D12, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, C24.3C22, C2×C4⋊Dic3, C2×D6⋊C4, C2×C4×C12, C22×D12, (C2×C4)⋊6D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, D6⋊C4, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C24.3C22, C4×D12, C4⋊D12, C42⋊7S3, C2×D6⋊C4, C12⋊7D4, (C2×C4)⋊6D12
►On 96 points
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)
(1 38 93 15)(2 39 94 16)(3 40 95 17)(4 41 96 18)(5 42 85 19)(6 43 86 20)(7 44 87 21)(8 45 88 22)(9 46 89 23)(10 47 90 24)(11 48 91 13)(12 37 92 14)(25 67 83 58)(26 68 84 59)(27 69 73 60)(28 70 74 49)(29 71 75 50)(30 72 76 51)(31 61 77 52)(32 62 78 53)(33 63 79 54)(34 64 80 55)(35 65 81 56)(36 66 82 57)
(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 75)(2 74)(3 73)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(14 24)(15 23)(16 22)(17 21)(18 20)(25 85)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 47)(38 46)(39 45)(40 44)(41 43)(49 55)(50 54)(51 53)(56 60)(57 59)(62 72)(63 71)(64 70)(65 69)(66 68)
G:=sub<Sym(96)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52), (1,38,93,15)(2,39,94,16)(3,40,95,17)(4,41,96,18)(5,42,85,19)(6,43,86,20)(7,44,87,21)(8,45,88,22)(9,46,89,23)(10,47,90,24)(11,48,91,13)(12,37,92,14)(25,67,83,58)(26,68,84,59)(27,69,73,60)(28,70,74,49)(29,71,75,50)(30,72,76,51)(31,61,77,52)(32,62,78,53)(33,63,79,54)(34,64,80,55)(35,65,81,56)(36,66,82,57), (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), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(14,24)(15,23)(16,22)(17,21)(18,20)(25,85)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,47)(38,46)(39,45)(40,44)(41,43)(49,55)(50,54)(51,53)(56,60)(57,59)(62,72)(63,71)(64,70)(65,69)(66,68)>;
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,49)(46,50)(47,51)(48,52), (1,38,93,15)(2,39,94,16)(3,40,95,17)(4,41,96,18)(5,42,85,19)(6,43,86,20)(7,44,87,21)(8,45,88,22)(9,46,89,23)(10,47,90,24)(11,48,91,13)(12,37,92,14)(25,67,83,58)(26,68,84,59)(27,69,73,60)(28,70,74,49)(29,71,75,50)(30,72,76,51)(31,61,77,52)(32,62,78,53)(33,63,79,54)(34,64,80,55)(35,65,81,56)(36,66,82,57), (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), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(14,24)(15,23)(16,22)(17,21)(18,20)(25,85)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,47)(38,46)(39,45)(40,44)(41,43)(49,55)(50,54)(51,53)(56,60)(57,59)(62,72)(63,71)(64,70)(65,69)(66,68) );
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,49),(46,50),(47,51),(48,52)], [(1,38,93,15),(2,39,94,16),(3,40,95,17),(4,41,96,18),(5,42,85,19),(6,43,86,20),(7,44,87,21),(8,45,88,22),(9,46,89,23),(10,47,90,24),(11,48,91,13),(12,37,92,14),(25,67,83,58),(26,68,84,59),(27,69,73,60),(28,70,74,49),(29,71,75,50),(30,72,76,51),(31,61,77,52),(32,62,78,53),(33,63,79,54),(34,64,80,55),(35,65,81,56),(36,66,82,57)], [(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)], [(1,75),(2,74),(3,73),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(14,24),(15,23),(16,22),(17,21),(18,20),(25,85),(26,96),(27,95),(28,94),(29,93),(30,92),(31,91),(32,90),(33,89),(34,88),(35,87),(36,86),(37,47),(38,46),(39,45),(40,44),(41,43),(49,55),(50,54),(51,53),(56,60),(57,59),(62,72),(63,71),(64,70),(65,69),(66,68)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | C4○D12 |
| kernel | (C2×C4)⋊6D12 | C2×C4⋊Dic3 | C2×D6⋊C4 | C2×C4×C12 | C22×D12 | C2×D12 | C2×C42 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 8 | 3 | 4 | 4 | 12 | 4 | 8 |
Matrix representation of (C2×C4)⋊6D12 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,5,0,0,0,0,3,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
(C2×C4)⋊6D12 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_6D_{12} % in TeX
G:=Group("(C2xC4):6D12"); // GroupNames label
G:=SmallGroup(192,498);
// by ID
G=gap.SmallGroup(192,498);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,758,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*c*d=c^-1>;
// generators/relations