metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).290D4, (C22×C4).60D6, (C22×S3).4Q8, C22.50(S3×Q8), C2.7(C12⋊3D4), C6.16(C4⋊1D4), C2.8(D6⋊3Q8), (C2×Dic3).61D4, C22.249(S3×D4), C6.50(C22⋊Q8), C3⋊3(C23.4Q8), C6.C42⋊43C2, C2.22(D6⋊Q8), C2.22(D6.D4), (S3×C23).22C22, (C22×C6).357C23, C23.392(C22×S3), (C22×C12).31C22, C22.110(C4○D12), C22.53(Q8⋊3S3), C6.54(C22.D4), C2.15(C23.28D6), (C22×Dic3).62C22, (C6×C4⋊C4)⋊27C2, (C2×C4⋊C4)⋊11S3, (C2×C6).85(C2×Q8), (C2×D6⋊C4).15C2, (C2×C6).453(C2×D4), (C2×Dic3⋊C4)⋊14C2, (C2×C4).43(C3⋊D4), (C2×C6).192(C4○D4), C22.142(C2×C3⋊D4), SmallGroup(192,552)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12).290D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab5, dcd=ab6c-1 >
Subgroups: 552 in 186 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×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, D6⋊C4, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.4Q8, C6.C42, C2×Dic3⋊C4, C2×D6⋊C4, C2×D6⋊C4, C6×C4⋊C4, (C2×C12).290D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C22.D4, C4⋊1D4, C4○D12, S3×D4, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.4Q8, D6.D4, D6⋊Q8, C23.28D6, C12⋊3D4, D6⋊3Q8, (C2×C12).290D4
►On 96 points
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 92)(26 93)(27 94)(28 95)(29 96)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(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 30 74 71)(2 96 75 49)(3 28 76 69)(4 94 77 59)(5 26 78 67)(6 92 79 57)(7 36 80 65)(8 90 81 55)(9 34 82 63)(10 88 83 53)(11 32 84 61)(12 86 73 51)(13 33 48 62)(14 87 37 52)(15 31 38 72)(16 85 39 50)(17 29 40 70)(18 95 41 60)(19 27 42 68)(20 93 43 58)(21 25 44 66)(22 91 45 56)(23 35 46 64)(24 89 47 54)
(2 21)(3 11)(4 19)(5 9)(6 17)(8 15)(10 13)(12 23)(14 18)(20 24)(25 64)(26 60)(27 62)(28 58)(29 72)(30 56)(31 70)(32 54)(33 68)(34 52)(35 66)(36 50)(37 41)(38 81)(40 79)(42 77)(43 47)(44 75)(46 73)(48 83)(49 86)(51 96)(53 94)(55 92)(57 90)(59 88)(61 89)(63 87)(65 85)(67 95)(69 93)(71 91)(76 84)(78 82)
G:=sub<Sym(96)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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), (1,30,74,71)(2,96,75,49)(3,28,76,69)(4,94,77,59)(5,26,78,67)(6,92,79,57)(7,36,80,65)(8,90,81,55)(9,34,82,63)(10,88,83,53)(11,32,84,61)(12,86,73,51)(13,33,48,62)(14,87,37,52)(15,31,38,72)(16,85,39,50)(17,29,40,70)(18,95,41,60)(19,27,42,68)(20,93,43,58)(21,25,44,66)(22,91,45,56)(23,35,46,64)(24,89,47,54), (2,21)(3,11)(4,19)(5,9)(6,17)(8,15)(10,13)(12,23)(14,18)(20,24)(25,64)(26,60)(27,62)(28,58)(29,72)(30,56)(31,70)(32,54)(33,68)(34,52)(35,66)(36,50)(37,41)(38,81)(40,79)(42,77)(43,47)(44,75)(46,73)(48,83)(49,86)(51,96)(53,94)(55,92)(57,90)(59,88)(61,89)(63,87)(65,85)(67,95)(69,93)(71,91)(76,84)(78,82)>;
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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), (1,30,74,71)(2,96,75,49)(3,28,76,69)(4,94,77,59)(5,26,78,67)(6,92,79,57)(7,36,80,65)(8,90,81,55)(9,34,82,63)(10,88,83,53)(11,32,84,61)(12,86,73,51)(13,33,48,62)(14,87,37,52)(15,31,38,72)(16,85,39,50)(17,29,40,70)(18,95,41,60)(19,27,42,68)(20,93,43,58)(21,25,44,66)(22,91,45,56)(23,35,46,64)(24,89,47,54), (2,21)(3,11)(4,19)(5,9)(6,17)(8,15)(10,13)(12,23)(14,18)(20,24)(25,64)(26,60)(27,62)(28,58)(29,72)(30,56)(31,70)(32,54)(33,68)(34,52)(35,66)(36,50)(37,41)(38,81)(40,79)(42,77)(43,47)(44,75)(46,73)(48,83)(49,86)(51,96)(53,94)(55,92)(57,90)(59,88)(61,89)(63,87)(65,85)(67,95)(69,93)(71,91)(76,84)(78,82) );
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,92),(26,93),(27,94),(28,95),(29,96),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,84),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(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)], [(1,30,74,71),(2,96,75,49),(3,28,76,69),(4,94,77,59),(5,26,78,67),(6,92,79,57),(7,36,80,65),(8,90,81,55),(9,34,82,63),(10,88,83,53),(11,32,84,61),(12,86,73,51),(13,33,48,62),(14,87,37,52),(15,31,38,72),(16,85,39,50),(17,29,40,70),(18,95,41,60),(19,27,42,68),(20,93,43,58),(21,25,44,66),(22,91,45,56),(23,35,46,64),(24,89,47,54)], [(2,21),(3,11),(4,19),(5,9),(6,17),(8,15),(10,13),(12,23),(14,18),(20,24),(25,64),(26,60),(27,62),(28,58),(29,72),(30,56),(31,70),(32,54),(33,68),(34,52),(35,66),(36,50),(37,41),(38,81),(40,79),(42,77),(43,47),(44,75),(46,73),(48,83),(49,86),(51,96),(53,94),(55,92),(57,90),(59,88),(61,89),(63,87),(65,85),(67,95),(69,93),(71,91),(76,84),(78,82)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | Q8 | D6 | C4○D4 | C3⋊D4 | C4○D12 | S3×D4 | S3×Q8 | Q8⋊3S3 |
| kernel | (C2×C12).290D4 | C6.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 4 | 2 | 2 | 3 | 6 | 4 | 8 | 2 | 1 | 1 |
Matrix representation of (C2×C12).290D4 ►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 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
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],[4,11,0,0,0,0,2,2,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[10,10,0,0,0,0,7,3,0,0,0,0,0,0,5,6,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,9,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
(C2×C12).290D4 in GAP, Magma, Sage, TeX
(C_2\times C_{12})._{290}D_4 % in TeX
G:=Group("(C2xC12).290D4"); // GroupNames label
G:=SmallGroup(192,552);
// by ID
G=gap.SmallGroup(192,552);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,387,268,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^5,d*c*d=a*b^6*c^-1>;
// generators/relations