metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.97D6, C6.982+ 1+4, C12⋊Q8⋊12C2, Dic3⋊D4⋊4C2, C4⋊C4.311D6, C12⋊D4⋊13C2, C12⋊7D4⋊30C2, C42⋊7S3⋊7C2, C42⋊2S3⋊4C2, (C2×C6).76C24, Dic3⋊5D4⋊13C2, C42⋊C2⋊16S3, C2.10(D4○D12), C4.97(C4○D12), (C4×C12).27C22, D6⋊C4.83C22, C22⋊C4.100D6, (C22×C4).213D6, C12.199(C4○D4), (C2×C12).697C23, C23.11D6⋊4C2, (C2×D12).24C22, C23.97(C22×S3), Dic3.18(C4○D4), (C22×S3).24C23, C4⋊Dic3.196C22, (C22×C6).146C23, C22.105(S3×C23), (C2×Dic3).29C23, (C2×Dic6).24C22, Dic3⋊C4.106C22, (C22×C12).233C22, C3⋊1(C22.49C24), (C4×Dic3).198C22, C6.D4.98C22, (C4×C3⋊D4)⋊13C2, C4⋊C4⋊7S3⋊12C2, C6.32(C2×C4○D4), C2.15(S3×C4○D4), C2.35(C2×C4○D12), (S3×C2×C4).194C22, (C3×C42⋊C2)⋊18C2, (C3×C4⋊C4).312C22, (C2×C4).278(C22×S3), (C2×C3⋊D4).105C22, (C3×C22⋊C4).115C22, SmallGroup(192,1091)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.97D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c5 >
Subgroups: 632 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C42⋊C2, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22.49C24, C42⋊2S3, C42⋊7S3, Dic3⋊D4, C23.11D6, C12⋊Q8, C4⋊C4⋊7S3, Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C12⋊7D4, C3×C42⋊C2, C42.97D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.49C24, C2×C4○D12, S3×C4○D4, D4○D12, C42.97D6
►On 96 points
(1 54 61 34)(2 49 62 29)(3 56 63 36)(4 51 64 31)(5 58 65 26)(6 53 66 33)(7 60 67 28)(8 55 68 35)(9 50 69 30)(10 57 70 25)(11 52 71 32)(12 59 72 27)(13 79 86 42)(14 74 87 37)(15 81 88 44)(16 76 89 39)(17 83 90 46)(18 78 91 41)(19 73 92 48)(20 80 93 43)(21 75 94 38)(22 82 95 45)(23 77 96 40)(24 84 85 47)
(1 75 7 81)(2 76 8 82)(3 77 9 83)(4 78 10 84)(5 79 11 73)(6 80 12 74)(13 32 19 26)(14 33 20 27)(15 34 21 28)(16 35 22 29)(17 36 23 30)(18 25 24 31)(37 66 43 72)(38 67 44 61)(39 68 45 62)(40 69 46 63)(41 70 47 64)(42 71 48 65)(49 89 55 95)(50 90 56 96)(51 91 57 85)(52 92 58 86)(53 93 59 87)(54 94 60 88)
(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 23 7 17)(2 16 8 22)(3 21 9 15)(4 14 10 20)(5 19 11 13)(6 24 12 18)(25 43 31 37)(26 48 32 42)(27 41 33 47)(28 46 34 40)(29 39 35 45)(30 44 36 38)(49 76 55 82)(50 81 56 75)(51 74 57 80)(52 79 58 73)(53 84 59 78)(54 77 60 83)(61 96 67 90)(62 89 68 95)(63 94 69 88)(64 87 70 93)(65 92 71 86)(66 85 72 91)
G:=sub<Sym(96)| (1,54,61,34)(2,49,62,29)(3,56,63,36)(4,51,64,31)(5,58,65,26)(6,53,66,33)(7,60,67,28)(8,55,68,35)(9,50,69,30)(10,57,70,25)(11,52,71,32)(12,59,72,27)(13,79,86,42)(14,74,87,37)(15,81,88,44)(16,76,89,39)(17,83,90,46)(18,78,91,41)(19,73,92,48)(20,80,93,43)(21,75,94,38)(22,82,95,45)(23,77,96,40)(24,84,85,47), (1,75,7,81)(2,76,8,82)(3,77,9,83)(4,78,10,84)(5,79,11,73)(6,80,12,74)(13,32,19,26)(14,33,20,27)(15,34,21,28)(16,35,22,29)(17,36,23,30)(18,25,24,31)(37,66,43,72)(38,67,44,61)(39,68,45,62)(40,69,46,63)(41,70,47,64)(42,71,48,65)(49,89,55,95)(50,90,56,96)(51,91,57,85)(52,92,58,86)(53,93,59,87)(54,94,60,88), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,43,31,37)(26,48,32,42)(27,41,33,47)(28,46,34,40)(29,39,35,45)(30,44,36,38)(49,76,55,82)(50,81,56,75)(51,74,57,80)(52,79,58,73)(53,84,59,78)(54,77,60,83)(61,96,67,90)(62,89,68,95)(63,94,69,88)(64,87,70,93)(65,92,71,86)(66,85,72,91)>;
G:=Group( (1,54,61,34)(2,49,62,29)(3,56,63,36)(4,51,64,31)(5,58,65,26)(6,53,66,33)(7,60,67,28)(8,55,68,35)(9,50,69,30)(10,57,70,25)(11,52,71,32)(12,59,72,27)(13,79,86,42)(14,74,87,37)(15,81,88,44)(16,76,89,39)(17,83,90,46)(18,78,91,41)(19,73,92,48)(20,80,93,43)(21,75,94,38)(22,82,95,45)(23,77,96,40)(24,84,85,47), (1,75,7,81)(2,76,8,82)(3,77,9,83)(4,78,10,84)(5,79,11,73)(6,80,12,74)(13,32,19,26)(14,33,20,27)(15,34,21,28)(16,35,22,29)(17,36,23,30)(18,25,24,31)(37,66,43,72)(38,67,44,61)(39,68,45,62)(40,69,46,63)(41,70,47,64)(42,71,48,65)(49,89,55,95)(50,90,56,96)(51,91,57,85)(52,92,58,86)(53,93,59,87)(54,94,60,88), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,43,31,37)(26,48,32,42)(27,41,33,47)(28,46,34,40)(29,39,35,45)(30,44,36,38)(49,76,55,82)(50,81,56,75)(51,74,57,80)(52,79,58,73)(53,84,59,78)(54,77,60,83)(61,96,67,90)(62,89,68,95)(63,94,69,88)(64,87,70,93)(65,92,71,86)(66,85,72,91) );
G=PermutationGroup([[(1,54,61,34),(2,49,62,29),(3,56,63,36),(4,51,64,31),(5,58,65,26),(6,53,66,33),(7,60,67,28),(8,55,68,35),(9,50,69,30),(10,57,70,25),(11,52,71,32),(12,59,72,27),(13,79,86,42),(14,74,87,37),(15,81,88,44),(16,76,89,39),(17,83,90,46),(18,78,91,41),(19,73,92,48),(20,80,93,43),(21,75,94,38),(22,82,95,45),(23,77,96,40),(24,84,85,47)], [(1,75,7,81),(2,76,8,82),(3,77,9,83),(4,78,10,84),(5,79,11,73),(6,80,12,74),(13,32,19,26),(14,33,20,27),(15,34,21,28),(16,35,22,29),(17,36,23,30),(18,25,24,31),(37,66,43,72),(38,67,44,61),(39,68,45,62),(40,69,46,63),(41,70,47,64),(42,71,48,65),(49,89,55,95),(50,90,56,96),(51,91,57,85),(52,92,58,86),(53,93,59,87),(54,94,60,88)], [(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)], [(1,23,7,17),(2,16,8,22),(3,21,9,15),(4,14,10,20),(5,19,11,13),(6,24,12,18),(25,43,31,37),(26,48,32,42),(27,41,33,47),(28,46,34,40),(29,39,35,45),(30,44,36,38),(49,76,55,82),(50,81,56,75),(51,74,57,80),(52,79,58,73),(53,84,59,78),(54,77,60,83),(61,96,67,90),(62,89,68,95),(63,94,69,88),(64,87,70,93),(65,92,71,86),(66,85,72,91)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ 1+4 | S3×C4○D4 | D4○D12 |
| kernel | C42.97D6 | C42⋊2S3 | C42⋊7S3 | Dic3⋊D4 | C23.11D6 | C12⋊Q8 | C4⋊C4⋊7S3 | Dic3⋊5D4 | C12⋊D4 | C4×C3⋊D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.97D6 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C42.97D6 in GAP, Magma, Sage, TeX
C_4^2._{97}D_6 % in TeX
G:=Group("C4^2.97D6"); // GroupNames label
G:=SmallGroup(192,1091);
// by ID
G=gap.SmallGroup(192,1091);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations