metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.233D6, (C4xS3):6D4, D6.6(C2xD4), C4.30(S3xD4), (S3xC42):8C2, C12.59(C2xD4), Dic3:D4:37C2, C12:3D4:23C2, (C2xD4).168D6, C4.4D4:18S3, (C2xQ8).158D6, C22:C4.70D6, C6.86(C22xD4), C42:7S3:22C2, Dic3:3(C4oD4), (C2xC12).76C23, (C2xC6).212C24, D6:C4.58C22, Dic3.65(C2xD4), Dic3:4D4:27C2, Dic3:Q8:19C2, (C4xC12).182C22, (C6xD4).150C22, (C22xC6).42C23, C23.44(C22xS3), (C6xQ8).121C22, (C2xD12).160C22, Dic3:C4.47C22, C22.233(S3xC23), C3:4(C22.26C24), (C22xS3).212C23, (C2xDic6).173C22, (C4xDic3).296C22, (C2xDic3).249C23, (C22xDic3).137C22, C2.59(C2xS3xD4), C2.71(S3xC4oD4), (C2xQ8:3S3):9C2, C6.183(C2xC4oD4), (C3xC4.4D4):6C2, (C2xD4:2S3):18C2, (S3xC2xC4).118C22, (C2xC4).298(C22xS3), (C2xC3:D4).55C22, (C3xC22:C4).59C22, SmallGroup(192,1227)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.233D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c-1 >
Subgroups: 816 in 310 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic6, C4xS3, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C2xC42, C4xD4, C4:D4, C4.4D4, C4.4D4, C4:1D4, C4:Q8, C2xC4oD4, C4xDic3, C4xDic3, Dic3:C4, D6:C4, C4xC12, C3xC22:C4, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, D4:2S3, Q8:3S3, C22xDic3, C2xC3:D4, C6xD4, C6xQ8, C22.26C24, S3xC42, C42:7S3, Dic3:4D4, Dic3:D4, C12:3D4, Dic3:Q8, C3xC4.4D4, C2xD4:2S3, C2xQ8:3S3, C42.233D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, S3xD4, S3xC23, C22.26C24, C2xS3xD4, S3xC4oD4, C42.233D6
►On 96 points
(1 80 19 66)(2 68 20 53)(3 82 21 62)(4 70 22 49)(5 84 23 64)(6 72 24 51)(7 79 25 65)(8 67 26 52)(9 81 27 61)(10 69 28 54)(11 83 29 63)(12 71 30 50)(13 47 86 38)(14 58 87 34)(15 43 88 40)(16 60 89 36)(17 45 90 42)(18 56 85 32)(31 93 55 73)(33 95 57 75)(35 91 59 77)(37 94 46 74)(39 96 48 76)(41 92 44 78)
(1 59 8 43)(2 36 9 41)(3 55 10 45)(4 32 11 37)(5 57 12 47)(6 34 7 39)(13 64 95 50)(14 79 96 72)(15 66 91 52)(16 81 92 68)(17 62 93 54)(18 83 94 70)(19 35 26 40)(20 60 27 44)(21 31 28 42)(22 56 29 46)(23 33 30 38)(24 58 25 48)(49 85 63 74)(51 87 65 76)(53 89 61 78)(67 88 80 77)(69 90 82 73)(71 86 84 75)
(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 73)(2 92)(3 77)(4 96)(5 75)(6 94)(7 18)(8 90)(9 16)(10 88)(11 14)(12 86)(13 30)(15 28)(17 26)(19 93)(20 78)(21 91)(22 76)(23 95)(24 74)(25 85)(27 89)(29 87)(31 67)(32 51)(33 71)(34 49)(35 69)(36 53)(37 65)(38 84)(39 63)(40 82)(41 61)(42 80)(43 62)(44 81)(45 66)(46 79)(47 64)(48 83)(50 57)(52 55)(54 59)(56 72)(58 70)(60 68)
G:=sub<Sym(96)| (1,80,19,66)(2,68,20,53)(3,82,21,62)(4,70,22,49)(5,84,23,64)(6,72,24,51)(7,79,25,65)(8,67,26,52)(9,81,27,61)(10,69,28,54)(11,83,29,63)(12,71,30,50)(13,47,86,38)(14,58,87,34)(15,43,88,40)(16,60,89,36)(17,45,90,42)(18,56,85,32)(31,93,55,73)(33,95,57,75)(35,91,59,77)(37,94,46,74)(39,96,48,76)(41,92,44,78), (1,59,8,43)(2,36,9,41)(3,55,10,45)(4,32,11,37)(5,57,12,47)(6,34,7,39)(13,64,95,50)(14,79,96,72)(15,66,91,52)(16,81,92,68)(17,62,93,54)(18,83,94,70)(19,35,26,40)(20,60,27,44)(21,31,28,42)(22,56,29,46)(23,33,30,38)(24,58,25,48)(49,85,63,74)(51,87,65,76)(53,89,61,78)(67,88,80,77)(69,90,82,73)(71,86,84,75), (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), (1,73)(2,92)(3,77)(4,96)(5,75)(6,94)(7,18)(8,90)(9,16)(10,88)(11,14)(12,86)(13,30)(15,28)(17,26)(19,93)(20,78)(21,91)(22,76)(23,95)(24,74)(25,85)(27,89)(29,87)(31,67)(32,51)(33,71)(34,49)(35,69)(36,53)(37,65)(38,84)(39,63)(40,82)(41,61)(42,80)(43,62)(44,81)(45,66)(46,79)(47,64)(48,83)(50,57)(52,55)(54,59)(56,72)(58,70)(60,68)>;
G:=Group( (1,80,19,66)(2,68,20,53)(3,82,21,62)(4,70,22,49)(5,84,23,64)(6,72,24,51)(7,79,25,65)(8,67,26,52)(9,81,27,61)(10,69,28,54)(11,83,29,63)(12,71,30,50)(13,47,86,38)(14,58,87,34)(15,43,88,40)(16,60,89,36)(17,45,90,42)(18,56,85,32)(31,93,55,73)(33,95,57,75)(35,91,59,77)(37,94,46,74)(39,96,48,76)(41,92,44,78), (1,59,8,43)(2,36,9,41)(3,55,10,45)(4,32,11,37)(5,57,12,47)(6,34,7,39)(13,64,95,50)(14,79,96,72)(15,66,91,52)(16,81,92,68)(17,62,93,54)(18,83,94,70)(19,35,26,40)(20,60,27,44)(21,31,28,42)(22,56,29,46)(23,33,30,38)(24,58,25,48)(49,85,63,74)(51,87,65,76)(53,89,61,78)(67,88,80,77)(69,90,82,73)(71,86,84,75), (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), (1,73)(2,92)(3,77)(4,96)(5,75)(6,94)(7,18)(8,90)(9,16)(10,88)(11,14)(12,86)(13,30)(15,28)(17,26)(19,93)(20,78)(21,91)(22,76)(23,95)(24,74)(25,85)(27,89)(29,87)(31,67)(32,51)(33,71)(34,49)(35,69)(36,53)(37,65)(38,84)(39,63)(40,82)(41,61)(42,80)(43,62)(44,81)(45,66)(46,79)(47,64)(48,83)(50,57)(52,55)(54,59)(56,72)(58,70)(60,68) );
G=PermutationGroup([[(1,80,19,66),(2,68,20,53),(3,82,21,62),(4,70,22,49),(5,84,23,64),(6,72,24,51),(7,79,25,65),(8,67,26,52),(9,81,27,61),(10,69,28,54),(11,83,29,63),(12,71,30,50),(13,47,86,38),(14,58,87,34),(15,43,88,40),(16,60,89,36),(17,45,90,42),(18,56,85,32),(31,93,55,73),(33,95,57,75),(35,91,59,77),(37,94,46,74),(39,96,48,76),(41,92,44,78)], [(1,59,8,43),(2,36,9,41),(3,55,10,45),(4,32,11,37),(5,57,12,47),(6,34,7,39),(13,64,95,50),(14,79,96,72),(15,66,91,52),(16,81,92,68),(17,62,93,54),(18,83,94,70),(19,35,26,40),(20,60,27,44),(21,31,28,42),(22,56,29,46),(23,33,30,38),(24,58,25,48),(49,85,63,74),(51,87,65,76),(53,89,61,78),(67,88,80,77),(69,90,82,73),(71,86,84,75)], [(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)], [(1,73),(2,92),(3,77),(4,96),(5,75),(6,94),(7,18),(8,90),(9,16),(10,88),(11,14),(12,86),(13,30),(15,28),(17,26),(19,93),(20,78),(21,91),(22,76),(23,95),(24,74),(25,85),(27,89),(29,87),(31,67),(32,51),(33,71),(34,49),(35,69),(36,53),(37,65),(38,84),(39,63),(40,82),(41,61),(42,80),(43,62),(44,81),(45,66),(46,79),(47,64),(48,83),(50,57),(52,55),(54,59),(56,72),(58,70),(60,68)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4oD4 | S3xD4 | S3xC4oD4 |
| kernel | C42.233D6 | S3xC42 | C42:7S3 | Dic3:4D4 | Dic3:D4 | C12:3D4 | Dic3:Q8 | C3xC4.4D4 | C2xD4:2S3 | C2xQ8:3S3 | C4.4D4 | C4xS3 | C42 | C22:C4 | C2xD4 | C2xQ8 | Dic3 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 8 | 2 | 4 |
Matrix representation of C42.233D6 ►in GL6(F13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 1 |
| 0 | 0 | 0 | 0 | 2 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,5,2,0,0,0,0,1,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,8,12],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,5,1] >;
C42.233D6 in GAP, Magma, Sage, TeX
C_4^2._{233}D_6 % in TeX
G:=Group("C4^2.233D6"); // GroupNames label
G:=SmallGroup(192,1227);
// by ID
G=gap.SmallGroup(192,1227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,232,100,1123,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^-1>;
// generators/relations