metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).56D4, (C2×C4).45D12, (C22×S3).5Q8, C22.51(S3×Q8), C6.63(C4⋊D4), C2.9(D6⋊3Q8), (C2×Dic3).62D4, (C22×C4).122D6, C22.250(S3×D4), C6.51(C22⋊Q8), C2.26(C12⋊D4), C2.11(C12⋊7D4), C3⋊3(C23.Q8), C6.C42⋊21C2, C2.23(D6⋊Q8), C2.17(C4.D12), C22.130(C2×D12), C6.30(C42⋊2C2), (S3×C23).23C22, C23.393(C22×S3), (C22×C6).358C23, (C22×C12).69C22, C2.13(C23.14D6), C22.111(C4○D12), C22.54(Q8⋊3S3), C22.105(D4⋊2S3), (C22×Dic3).63C22, (C6×C4⋊C4)⋊23C2, (C2×C4⋊C4)⋊12S3, (C2×C6).86(C2×Q8), (C2×D6⋊C4).24C2, (C2×C4⋊Dic3)⋊14C2, (C2×C6).338(C2×D4), (C2×Dic3⋊C4)⋊42C2, (C2×C6).87(C4○D4), (C2×C4).44(C3⋊D4), C2.15(C4⋊C4⋊S3), C22.143(C2×C3⋊D4), SmallGroup(192,553)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12).56D4
G = < a,b,c,d | a2=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b2c-1 >
Subgroups: 552 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, S3×C23, C23.Q8, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, C6×C4⋊C4, (C2×C12).56D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C22⋊Q8, C42⋊2C2, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.Q8, C12⋊D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C12⋊7D4, C23.14D6, D6⋊3Q8, (C2×C12).56D4
►On 96 points
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(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 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 85)(33 86)(34 87)(35 88)(36 89)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)
(1 22 80 91)(2 92 81 23)(3 24 82 93)(4 94 83 13)(5 14 84 95)(6 96 73 15)(7 16 74 85)(8 86 75 17)(9 18 76 87)(10 88 77 19)(11 20 78 89)(12 90 79 21)(25 55 69 45)(26 46 70 56)(27 57 71 47)(28 48 72 58)(29 59 61 37)(30 38 62 60)(31 49 63 39)(32 40 64 50)(33 51 65 41)(34 42 66 52)(35 53 67 43)(36 44 68 54)
(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 79 80 12)(2 11 81 78)(3 77 82 10)(4 9 83 76)(5 75 84 8)(6 7 73 74)(13 66 94 34)(14 33 95 65)(15 64 96 32)(16 31 85 63)(17 62 86 30)(18 29 87 61)(19 72 88 28)(20 27 89 71)(21 70 90 26)(22 25 91 69)(23 68 92 36)(24 35 93 67)(37 42 59 52)(38 51 60 41)(39 40 49 50)(43 48 53 58)(44 57 54 47)(45 46 55 56)
G:=sub<Sym(96)| (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84), (1,22,80,91)(2,92,81,23)(3,24,82,93)(4,94,83,13)(5,14,84,95)(6,96,73,15)(7,16,74,85)(8,86,75,17)(9,18,76,87)(10,88,77,19)(11,20,78,89)(12,90,79,21)(25,55,69,45)(26,46,70,56)(27,57,71,47)(28,48,72,58)(29,59,61,37)(30,38,62,60)(31,49,63,39)(32,40,64,50)(33,51,65,41)(34,42,66,52)(35,53,67,43)(36,44,68,54), (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), (1,79,80,12)(2,11,81,78)(3,77,82,10)(4,9,83,76)(5,75,84,8)(6,7,73,74)(13,66,94,34)(14,33,95,65)(15,64,96,32)(16,31,85,63)(17,62,86,30)(18,29,87,61)(19,72,88,28)(20,27,89,71)(21,70,90,26)(22,25,91,69)(23,68,92,36)(24,35,93,67)(37,42,59,52)(38,51,60,41)(39,40,49,50)(43,48,53,58)(44,57,54,47)(45,46,55,56)>;
G:=Group( (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,85)(33,86)(34,87)(35,88)(36,89)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84), (1,22,80,91)(2,92,81,23)(3,24,82,93)(4,94,83,13)(5,14,84,95)(6,96,73,15)(7,16,74,85)(8,86,75,17)(9,18,76,87)(10,88,77,19)(11,20,78,89)(12,90,79,21)(25,55,69,45)(26,46,70,56)(27,57,71,47)(28,48,72,58)(29,59,61,37)(30,38,62,60)(31,49,63,39)(32,40,64,50)(33,51,65,41)(34,42,66,52)(35,53,67,43)(36,44,68,54), (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), (1,79,80,12)(2,11,81,78)(3,77,82,10)(4,9,83,76)(5,75,84,8)(6,7,73,74)(13,66,94,34)(14,33,95,65)(15,64,96,32)(16,31,85,63)(17,62,86,30)(18,29,87,61)(19,72,88,28)(20,27,89,71)(21,70,90,26)(22,25,91,69)(23,68,92,36)(24,35,93,67)(37,42,59,52)(38,51,60,41)(39,40,49,50)(43,48,53,58)(44,57,54,47)(45,46,55,56) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,85),(33,86),(34,87),(35,88),(36,89),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84)], [(1,22,80,91),(2,92,81,23),(3,24,82,93),(4,94,83,13),(5,14,84,95),(6,96,73,15),(7,16,74,85),(8,86,75,17),(9,18,76,87),(10,88,77,19),(11,20,78,89),(12,90,79,21),(25,55,69,45),(26,46,70,56),(27,57,71,47),(28,48,72,58),(29,59,61,37),(30,38,62,60),(31,49,63,39),(32,40,64,50),(33,51,65,41),(34,42,66,52),(35,53,67,43),(36,44,68,54)], [(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)], [(1,79,80,12),(2,11,81,78),(3,77,82,10),(4,9,83,76),(5,75,84,8),(6,7,73,74),(13,66,94,34),(14,33,95,65),(15,64,96,32),(16,31,85,63),(17,62,86,30),(18,29,87,61),(19,72,88,28),(20,27,89,71),(21,70,90,26),(22,25,91,69),(23,68,92,36),(24,35,93,67),(37,42,59,52),(38,51,60,41),(39,40,49,50),(43,48,53,58),(44,57,54,47),(45,46,55,56)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | Q8 | D6 | C4○D4 | D12 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | (C2×C12).56D4 | C6.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 4 | 2 | 3 | 6 | 4 | 4 | 4 | 1 | 1 | 1 | 1 |
Matrix representation of (C2×C12).56D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[10,10,0,0,0,0,3,7,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,5,0,0,0,0,0,0,8] >;
(C2×C12).56D4 in GAP, Magma, Sage, TeX
(C_2\times C_{12})._{56}D_4 % in TeX
G:=Group("(C2xC12).56D4"); // GroupNames label
G:=SmallGroup(192,553);
// by ID
G=gap.SmallGroup(192,553);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,184,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations