metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6).40D8, C6.49(C2×D8), C4⋊C4.227D6, C6.D8⋊25C2, (C2×C12).283D4, C12⋊7D4.9C2, C6.Q16⋊25C2, C4.86(C4○D12), C12.55D4⋊4C2, C22.9(D4⋊S3), (C22×C6).185D4, (C22×C4).110D6, C3⋊4(C22.D8), C12.174(C4○D4), (C2×C12).320C23, (C2×D12).90C22, C6.84(C8.C22), C23.84(C3⋊D4), C2.6(Q8.11D6), C4⋊Dic3.130C22, (C22×C12).135C22, C6.58(C22.D4), C2.8(C23.28D6), (C6×C4⋊C4)⋊3C2, (C2×C4⋊C4)⋊3S3, C2.5(C2×D4⋊S3), (C2×C6).440(C2×D4), (C2×C3⋊C8).81C22, (C2×C4).31(C3⋊D4), (C3×C4⋊C4).258C22, (C2×C4).420(C22×S3), C22.130(C2×C3⋊D4), SmallGroup(192,526)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6).40D8
G = < a,b,c,d | a6=b2=c8=1, d2=a3, ab=ba, cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 344 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C22.D8, C6.Q16, C6.D8, C12.55D4, C12⋊7D4, C6×C4⋊C4, (C2×C6).40D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×D8, C8.C22, D4⋊S3, C4○D12, C2×C3⋊D4, C22.D8, C23.28D6, C2×D4⋊S3, Q8.11D6, (C2×C6).40D8
►On 96 points
(1 16 38 68 51 81)(2 82 52 69 39 9)(3 10 40 70 53 83)(4 84 54 71 33 11)(5 12 34 72 55 85)(6 86 56 65 35 13)(7 14 36 66 49 87)(8 88 50 67 37 15)(17 89 75 63 29 41)(18 42 30 64 76 90)(19 91 77 57 31 43)(20 44 32 58 78 92)(21 93 79 59 25 45)(22 46 26 60 80 94)(23 95 73 61 27 47)(24 48 28 62 74 96)
(1 18)(2 57)(3 20)(4 59)(5 22)(6 61)(7 24)(8 63)(9 77)(10 44)(11 79)(12 46)(13 73)(14 48)(15 75)(16 42)(17 67)(19 69)(21 71)(23 65)(25 84)(26 34)(27 86)(28 36)(29 88)(30 38)(31 82)(32 40)(33 93)(35 95)(37 89)(39 91)(41 50)(43 52)(45 54)(47 56)(49 74)(51 76)(53 78)(55 80)(58 70)(60 72)(62 66)(64 68)(81 90)(83 92)(85 94)(87 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 68 59)(2 20 69 58)(3 19 70 57)(4 18 71 64)(5 17 72 63)(6 24 65 62)(7 23 66 61)(8 22 67 60)(9 92 52 32)(10 91 53 31)(11 90 54 30)(12 89 55 29)(13 96 56 28)(14 95 49 27)(15 94 50 26)(16 93 51 25)(33 76 84 42)(34 75 85 41)(35 74 86 48)(36 73 87 47)(37 80 88 46)(38 79 81 45)(39 78 82 44)(40 77 83 43)
G:=sub<Sym(96)| (1,16,38,68,51,81)(2,82,52,69,39,9)(3,10,40,70,53,83)(4,84,54,71,33,11)(5,12,34,72,55,85)(6,86,56,65,35,13)(7,14,36,66,49,87)(8,88,50,67,37,15)(17,89,75,63,29,41)(18,42,30,64,76,90)(19,91,77,57,31,43)(20,44,32,58,78,92)(21,93,79,59,25,45)(22,46,26,60,80,94)(23,95,73,61,27,47)(24,48,28,62,74,96), (1,18)(2,57)(3,20)(4,59)(5,22)(6,61)(7,24)(8,63)(9,77)(10,44)(11,79)(12,46)(13,73)(14,48)(15,75)(16,42)(17,67)(19,69)(21,71)(23,65)(25,84)(26,34)(27,86)(28,36)(29,88)(30,38)(31,82)(32,40)(33,93)(35,95)(37,89)(39,91)(41,50)(43,52)(45,54)(47,56)(49,74)(51,76)(53,78)(55,80)(58,70)(60,72)(62,66)(64,68)(81,90)(83,92)(85,94)(87,96), (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), (1,21,68,59)(2,20,69,58)(3,19,70,57)(4,18,71,64)(5,17,72,63)(6,24,65,62)(7,23,66,61)(8,22,67,60)(9,92,52,32)(10,91,53,31)(11,90,54,30)(12,89,55,29)(13,96,56,28)(14,95,49,27)(15,94,50,26)(16,93,51,25)(33,76,84,42)(34,75,85,41)(35,74,86,48)(36,73,87,47)(37,80,88,46)(38,79,81,45)(39,78,82,44)(40,77,83,43)>;
G:=Group( (1,16,38,68,51,81)(2,82,52,69,39,9)(3,10,40,70,53,83)(4,84,54,71,33,11)(5,12,34,72,55,85)(6,86,56,65,35,13)(7,14,36,66,49,87)(8,88,50,67,37,15)(17,89,75,63,29,41)(18,42,30,64,76,90)(19,91,77,57,31,43)(20,44,32,58,78,92)(21,93,79,59,25,45)(22,46,26,60,80,94)(23,95,73,61,27,47)(24,48,28,62,74,96), (1,18)(2,57)(3,20)(4,59)(5,22)(6,61)(7,24)(8,63)(9,77)(10,44)(11,79)(12,46)(13,73)(14,48)(15,75)(16,42)(17,67)(19,69)(21,71)(23,65)(25,84)(26,34)(27,86)(28,36)(29,88)(30,38)(31,82)(32,40)(33,93)(35,95)(37,89)(39,91)(41,50)(43,52)(45,54)(47,56)(49,74)(51,76)(53,78)(55,80)(58,70)(60,72)(62,66)(64,68)(81,90)(83,92)(85,94)(87,96), (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), (1,21,68,59)(2,20,69,58)(3,19,70,57)(4,18,71,64)(5,17,72,63)(6,24,65,62)(7,23,66,61)(8,22,67,60)(9,92,52,32)(10,91,53,31)(11,90,54,30)(12,89,55,29)(13,96,56,28)(14,95,49,27)(15,94,50,26)(16,93,51,25)(33,76,84,42)(34,75,85,41)(35,74,86,48)(36,73,87,47)(37,80,88,46)(38,79,81,45)(39,78,82,44)(40,77,83,43) );
G=PermutationGroup([[(1,16,38,68,51,81),(2,82,52,69,39,9),(3,10,40,70,53,83),(4,84,54,71,33,11),(5,12,34,72,55,85),(6,86,56,65,35,13),(7,14,36,66,49,87),(8,88,50,67,37,15),(17,89,75,63,29,41),(18,42,30,64,76,90),(19,91,77,57,31,43),(20,44,32,58,78,92),(21,93,79,59,25,45),(22,46,26,60,80,94),(23,95,73,61,27,47),(24,48,28,62,74,96)], [(1,18),(2,57),(3,20),(4,59),(5,22),(6,61),(7,24),(8,63),(9,77),(10,44),(11,79),(12,46),(13,73),(14,48),(15,75),(16,42),(17,67),(19,69),(21,71),(23,65),(25,84),(26,34),(27,86),(28,36),(29,88),(30,38),(31,82),(32,40),(33,93),(35,95),(37,89),(39,91),(41,50),(43,52),(45,54),(47,56),(49,74),(51,76),(53,78),(55,80),(58,70),(60,72),(62,66),(64,68),(81,90),(83,92),(85,94),(87,96)], [(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)], [(1,21,68,59),(2,20,69,58),(3,19,70,57),(4,18,71,64),(5,17,72,63),(6,24,65,62),(7,23,66,61),(8,22,67,60),(9,92,52,32),(10,91,53,31),(11,90,54,30),(12,89,55,29),(13,96,56,28),(14,95,49,27),(15,94,50,26),(16,93,51,25),(33,76,84,42),(34,75,85,41),(35,74,86,48),(36,73,87,47),(37,80,88,46),(38,79,81,45),(39,78,82,44),(40,77,83,43)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D8 | C3⋊D4 | C3⋊D4 | C4○D12 | C8.C22 | D4⋊S3 | Q8.11D6 |
| kernel | (C2×C6).40D8 | C6.Q16 | C6.D8 | C12.55D4 | C12⋊7D4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C4 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of (C2×C6).40D8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
| 57 | 57 | 0 | 0 |
| 16 | 57 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 59 |
| 57 | 57 | 0 | 0 |
| 57 | 16 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[72,0,0,0,0,72,0,0,0,0,43,13,0,0,60,30],[57,16,0,0,57,57,0,0,0,0,14,7,0,0,66,59],[57,57,0,0,57,16,0,0,0,0,46,0,0,0,0,46] >;
(C2×C6).40D8 in GAP, Magma, Sage, TeX
(C_2\times C_6)._{40}D_8 % in TeX
G:=Group("(C2xC6).40D8"); // GroupNames label
G:=SmallGroup(192,526);
// by ID
G=gap.SmallGroup(192,526);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations