metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.157D6, C6.312- 1+4, C6.1352+ 1+4, C12⋊Q8⋊38C2, C4⋊C4.114D6, C4.D12⋊39C2, D6⋊Q8⋊37C2, C42.C2⋊13S3, C2.60(D4○D12), (C2×C6).243C24, D6⋊C4.43C22, C2.61(Q8○D12), C12.6Q8⋊30C2, D6.D4.4C2, (C4×C12).224C22, (C2×C12).190C23, C42⋊7S3.12C2, (C2×D12).36C22, Dic3⋊C4.86C22, C4⋊Dic3.245C22, C22.264(S3×C23), (C2×Dic6).41C22, (C22×S3).108C23, C2.32(Q8.15D6), C3⋊4(C22.57C24), (C2×Dic3).125C23, (C4×Dic3).148C22, C4⋊C4⋊S3⋊38C2, (S3×C2×C4).133C22, (C3×C42.C2)⋊16C2, (C3×C4⋊C4).198C22, (C2×C4).207(C22×S3), SmallGroup(192,1258)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.157D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 496 in 196 conjugacy classes, 91 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, C22.57C24, C12.6Q8, C42⋊7S3, C12⋊Q8, D6.D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.157D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, Q8.15D6, D4○D12, Q8○D12, C42.157D6
►On 96 points
Generators in S96
(1 56 34 40)(2 51 35 47)(3 58 36 42)(4 53 25 37)(5 60 26 44)(6 55 27 39)(7 50 28 46)(8 57 29 41)(9 52 30 48)(10 59 31 43)(11 54 32 38)(12 49 33 45)(13 65 86 82)(14 72 87 77)(15 67 88 84)(16 62 89 79)(17 69 90 74)(18 64 91 81)(19 71 92 76)(20 66 93 83)(21 61 94 78)(22 68 95 73)(23 63 96 80)(24 70 85 75)
(1 68 28 79)(2 80 29 69)(3 70 30 81)(4 82 31 71)(5 72 32 83)(6 84 33 61)(7 62 34 73)(8 74 35 63)(9 64 36 75)(10 76 25 65)(11 66 26 77)(12 78 27 67)(13 43 92 53)(14 54 93 44)(15 45 94 55)(16 56 95 46)(17 47 96 57)(18 58 85 48)(19 37 86 59)(20 60 87 38)(21 39 88 49)(22 50 89 40)(23 41 90 51)(24 52 91 42)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 52 43 58)(38 57 44 51)(39 50 45 56)(40 55 46 49)(41 60 47 54)(42 53 48 59)(61 79 67 73)(62 84 68 78)(63 77 69 83)(64 82 70 76)(65 75 71 81)(66 80 72 74)(85 92 91 86)(87 90 93 96)(88 95 94 89)
G:=sub<Sym(96)| (1,56,34,40)(2,51,35,47)(3,58,36,42)(4,53,25,37)(5,60,26,44)(6,55,27,39)(7,50,28,46)(8,57,29,41)(9,52,30,48)(10,59,31,43)(11,54,32,38)(12,49,33,45)(13,65,86,82)(14,72,87,77)(15,67,88,84)(16,62,89,79)(17,69,90,74)(18,64,91,81)(19,71,92,76)(20,66,93,83)(21,61,94,78)(22,68,95,73)(23,63,96,80)(24,70,85,75), (1,68,28,79)(2,80,29,69)(3,70,30,81)(4,82,31,71)(5,72,32,83)(6,84,33,61)(7,62,34,73)(8,74,35,63)(9,64,36,75)(10,76,25,65)(11,66,26,77)(12,78,27,67)(13,43,92,53)(14,54,93,44)(15,45,94,55)(16,56,95,46)(17,47,96,57)(18,58,85,48)(19,37,86,59)(20,60,87,38)(21,39,88,49)(22,50,89,40)(23,41,90,51)(24,52,91,42), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,52,43,58)(38,57,44,51)(39,50,45,56)(40,55,46,49)(41,60,47,54)(42,53,48,59)(61,79,67,73)(62,84,68,78)(63,77,69,83)(64,82,70,76)(65,75,71,81)(66,80,72,74)(85,92,91,86)(87,90,93,96)(88,95,94,89)>;
G:=Group( (1,56,34,40)(2,51,35,47)(3,58,36,42)(4,53,25,37)(5,60,26,44)(6,55,27,39)(7,50,28,46)(8,57,29,41)(9,52,30,48)(10,59,31,43)(11,54,32,38)(12,49,33,45)(13,65,86,82)(14,72,87,77)(15,67,88,84)(16,62,89,79)(17,69,90,74)(18,64,91,81)(19,71,92,76)(20,66,93,83)(21,61,94,78)(22,68,95,73)(23,63,96,80)(24,70,85,75), (1,68,28,79)(2,80,29,69)(3,70,30,81)(4,82,31,71)(5,72,32,83)(6,84,33,61)(7,62,34,73)(8,74,35,63)(9,64,36,75)(10,76,25,65)(11,66,26,77)(12,78,27,67)(13,43,92,53)(14,54,93,44)(15,45,94,55)(16,56,95,46)(17,47,96,57)(18,58,85,48)(19,37,86,59)(20,60,87,38)(21,39,88,49)(22,50,89,40)(23,41,90,51)(24,52,91,42), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,52,43,58)(38,57,44,51)(39,50,45,56)(40,55,46,49)(41,60,47,54)(42,53,48,59)(61,79,67,73)(62,84,68,78)(63,77,69,83)(64,82,70,76)(65,75,71,81)(66,80,72,74)(85,92,91,86)(87,90,93,96)(88,95,94,89) );
G=PermutationGroup([[(1,56,34,40),(2,51,35,47),(3,58,36,42),(4,53,25,37),(5,60,26,44),(6,55,27,39),(7,50,28,46),(8,57,29,41),(9,52,30,48),(10,59,31,43),(11,54,32,38),(12,49,33,45),(13,65,86,82),(14,72,87,77),(15,67,88,84),(16,62,89,79),(17,69,90,74),(18,64,91,81),(19,71,92,76),(20,66,93,83),(21,61,94,78),(22,68,95,73),(23,63,96,80),(24,70,85,75)], [(1,68,28,79),(2,80,29,69),(3,70,30,81),(4,82,31,71),(5,72,32,83),(6,84,33,61),(7,62,34,73),(8,74,35,63),(9,64,36,75),(10,76,25,65),(11,66,26,77),(12,78,27,67),(13,43,92,53),(14,54,93,44),(15,45,94,55),(16,56,95,46),(17,47,96,57),(18,58,85,48),(19,37,86,59),(20,60,87,38),(21,39,88,49),(22,50,89,40),(23,41,90,51),(24,52,91,42)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,52,43,58),(38,57,44,51),(39,50,45,56),(40,55,46,49),(41,60,47,54),(42,53,48,59),(61,79,67,73),(62,84,68,78),(63,77,69,83),(64,82,70,76),(65,75,71,81),(66,80,72,74),(85,92,91,86),(87,90,93,96),(88,95,94,89)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4G | 4H | ··· | 4M | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2+ 1+4 | 2- 1+4 | Q8.15D6 | D4○D12 | Q8○D12 |
| kernel | C42.157D6 | C12.6Q8 | C42⋊7S3 | C12⋊Q8 | D6.D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 6 | 1 | 2 | 2 | 2 | 2 |
Matrix representation of C42.157D6 ►in GL8(𝔽13)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 2 |
| 6 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 7 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 6 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 |
| 6 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 6 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 7 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
G:=sub<GL(8,GF(13))| [0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2],[6,0,0,10,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0],[6,0,0,10,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,3,0,0,7,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0,0,0,5,0,0,0,0,0,0,0,8,8,0,0] >;
C42.157D6 in GAP, Magma, Sage, TeX
C_4^2._{157}D_6 % in TeX
G:=Group("C4^2.157D6"); // GroupNames label
G:=SmallGroup(192,1258);
// by ID
G=gap.SmallGroup(192,1258);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations