metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.161D6, C6.1392+ 1+4, C6.1002- 1+4, (C4×D12)⋊15C2, C4⋊C4.118D6, C42⋊2C2⋊4S3, D6⋊Q8⋊43C2, C4.D12⋊41C2, Dic3⋊D4.4C2, C22⋊C4.79D6, C12.6Q8⋊9C2, D6.14(C4○D4), D6.D4⋊41C2, C23.9D6⋊51C2, C2.64(D4○D12), (C4×C12).33C22, (C2×C6).251C24, C2.64(Q8○D12), C4.Dic6⋊39C2, Dic3⋊4D4⋊36C2, (C2×C12).194C23, D6⋊C4.114C22, C4⋊Dic3.54C22, (C22×C6).65C23, C23.67(C22×S3), Dic3.D4⋊45C2, (C2×D12).227C22, C23.21D6⋊29C2, Dic3⋊C4.56C22, C22.272(S3×C23), (C22×S3).225C23, C3⋊9(C22.33C24), (C2×Dic3).265C23, (C2×Dic6).184C22, (C4×Dic3).151C22, C6.D4.67C22, (C22×Dic3).151C22, (S3×C4⋊C4)⋊41C2, C4⋊C4⋊S3⋊42C2, C2.98(S3×C4○D4), C6.209(C2×C4○D4), (C3×C42⋊2C2)⋊6C2, (S3×C2×C4).219C22, (C3×C4⋊C4).203C22, (C2×C4).209(C22×S3), (C2×C3⋊D4).71C22, (C3×C22⋊C4).76C22, SmallGroup(192,1266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.161D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 560 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22.33C24, C12.6Q8, C4×D12, Dic3.D4, Dic3⋊4D4, C23.9D6, Dic3⋊D4, C23.21D6, C4.Dic6, S3×C4⋊C4, D6.D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.161D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, S3×C23, C22.33C24, S3×C4○D4, D4○D12, Q8○D12, C42.161D6
►On 96 points
(1 55 37 83)(2 50 38 78)(3 57 39 73)(4 52 40 80)(5 59 41 75)(6 54 42 82)(7 49 43 77)(8 56 44 84)(9 51 45 79)(10 58 46 74)(11 53 47 81)(12 60 48 76)(13 62 35 87)(14 69 36 94)(15 64 25 89)(16 71 26 96)(17 66 27 91)(18 61 28 86)(19 68 29 93)(20 63 30 88)(21 70 31 95)(22 65 32 90)(23 72 33 85)(24 67 34 92)
(1 25 7 31)(2 22 8 16)(3 27 9 33)(4 24 10 18)(5 29 11 35)(6 14 12 20)(13 41 19 47)(15 43 21 37)(17 45 23 39)(26 38 32 44)(28 40 34 46)(30 42 36 48)(49 95 55 89)(50 65 56 71)(51 85 57 91)(52 67 58 61)(53 87 59 93)(54 69 60 63)(62 75 68 81)(64 77 70 83)(66 79 72 73)(74 86 80 92)(76 88 82 94)(78 90 84 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 36 7 30)(2 29 8 35)(3 34 9 28)(4 27 10 33)(5 32 11 26)(6 25 12 31)(13 38 19 44)(14 43 20 37)(15 48 21 42)(16 41 22 47)(17 46 23 40)(18 39 24 45)(49 94 55 88)(50 87 56 93)(51 92 57 86)(52 85 58 91)(53 90 59 96)(54 95 60 89)(61 79 67 73)(62 84 68 78)(63 77 69 83)(64 82 70 76)(65 75 71 81)(66 80 72 74)
G:=sub<Sym(96)| (1,55,37,83)(2,50,38,78)(3,57,39,73)(4,52,40,80)(5,59,41,75)(6,54,42,82)(7,49,43,77)(8,56,44,84)(9,51,45,79)(10,58,46,74)(11,53,47,81)(12,60,48,76)(13,62,35,87)(14,69,36,94)(15,64,25,89)(16,71,26,96)(17,66,27,91)(18,61,28,86)(19,68,29,93)(20,63,30,88)(21,70,31,95)(22,65,32,90)(23,72,33,85)(24,67,34,92), (1,25,7,31)(2,22,8,16)(3,27,9,33)(4,24,10,18)(5,29,11,35)(6,14,12,20)(13,41,19,47)(15,43,21,37)(17,45,23,39)(26,38,32,44)(28,40,34,46)(30,42,36,48)(49,95,55,89)(50,65,56,71)(51,85,57,91)(52,67,58,61)(53,87,59,93)(54,69,60,63)(62,75,68,81)(64,77,70,83)(66,79,72,73)(74,86,80,92)(76,88,82,94)(78,90,84,96), (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), (1,36,7,30)(2,29,8,35)(3,34,9,28)(4,27,10,33)(5,32,11,26)(6,25,12,31)(13,38,19,44)(14,43,20,37)(15,48,21,42)(16,41,22,47)(17,46,23,40)(18,39,24,45)(49,94,55,88)(50,87,56,93)(51,92,57,86)(52,85,58,91)(53,90,59,96)(54,95,60,89)(61,79,67,73)(62,84,68,78)(63,77,69,83)(64,82,70,76)(65,75,71,81)(66,80,72,74)>;
G:=Group( (1,55,37,83)(2,50,38,78)(3,57,39,73)(4,52,40,80)(5,59,41,75)(6,54,42,82)(7,49,43,77)(8,56,44,84)(9,51,45,79)(10,58,46,74)(11,53,47,81)(12,60,48,76)(13,62,35,87)(14,69,36,94)(15,64,25,89)(16,71,26,96)(17,66,27,91)(18,61,28,86)(19,68,29,93)(20,63,30,88)(21,70,31,95)(22,65,32,90)(23,72,33,85)(24,67,34,92), (1,25,7,31)(2,22,8,16)(3,27,9,33)(4,24,10,18)(5,29,11,35)(6,14,12,20)(13,41,19,47)(15,43,21,37)(17,45,23,39)(26,38,32,44)(28,40,34,46)(30,42,36,48)(49,95,55,89)(50,65,56,71)(51,85,57,91)(52,67,58,61)(53,87,59,93)(54,69,60,63)(62,75,68,81)(64,77,70,83)(66,79,72,73)(74,86,80,92)(76,88,82,94)(78,90,84,96), (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), (1,36,7,30)(2,29,8,35)(3,34,9,28)(4,27,10,33)(5,32,11,26)(6,25,12,31)(13,38,19,44)(14,43,20,37)(15,48,21,42)(16,41,22,47)(17,46,23,40)(18,39,24,45)(49,94,55,88)(50,87,56,93)(51,92,57,86)(52,85,58,91)(53,90,59,96)(54,95,60,89)(61,79,67,73)(62,84,68,78)(63,77,69,83)(64,82,70,76)(65,75,71,81)(66,80,72,74) );
G=PermutationGroup([[(1,55,37,83),(2,50,38,78),(3,57,39,73),(4,52,40,80),(5,59,41,75),(6,54,42,82),(7,49,43,77),(8,56,44,84),(9,51,45,79),(10,58,46,74),(11,53,47,81),(12,60,48,76),(13,62,35,87),(14,69,36,94),(15,64,25,89),(16,71,26,96),(17,66,27,91),(18,61,28,86),(19,68,29,93),(20,63,30,88),(21,70,31,95),(22,65,32,90),(23,72,33,85),(24,67,34,92)], [(1,25,7,31),(2,22,8,16),(3,27,9,33),(4,24,10,18),(5,29,11,35),(6,14,12,20),(13,41,19,47),(15,43,21,37),(17,45,23,39),(26,38,32,44),(28,40,34,46),(30,42,36,48),(49,95,55,89),(50,65,56,71),(51,85,57,91),(52,67,58,61),(53,87,59,93),(54,69,60,63),(62,75,68,81),(64,77,70,83),(66,79,72,73),(74,86,80,92),(76,88,82,94),(78,90,84,96)], [(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)], [(1,36,7,30),(2,29,8,35),(3,34,9,28),(4,27,10,33),(5,32,11,26),(6,25,12,31),(13,38,19,44),(14,43,20,37),(15,48,21,42),(16,41,22,47),(17,46,23,40),(18,39,24,45),(49,94,55,88),(50,87,56,93),(51,92,57,86),(52,85,58,91),(53,90,59,96),(54,95,60,89),(61,79,67,73),(62,84,68,78),(63,77,69,83),(64,82,70,76),(65,75,71,81),(66,80,72,74)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | S3×C4○D4 | D4○D12 | Q8○D12 |
| kernel | C42.161D6 | C12.6Q8 | C4×D12 | Dic3.D4 | Dic3⋊4D4 | C23.9D6 | Dic3⋊D4 | C23.21D6 | C4.Dic6 | S3×C4⋊C4 | D6.D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | D6 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.161D6 ►in GL8(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 8 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 8 | 0 |
G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,11,1,0],[1,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,5,8,0,0,0,0,0,5,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0] >;
C42.161D6 in GAP, Magma, Sage, TeX
C_4^2._{161}D_6 % in TeX
G:=Group("C4^2.161D6"); // GroupNames label
G:=SmallGroup(192,1266);
// by ID
G=gap.SmallGroup(192,1266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,570,80,6278]);
// 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=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations