metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.151D6, C6.292- 1+4, C4⋊C4.112D6, C42.C2⋊7S3, D6⋊Q8⋊36C2, C42⋊2S3⋊37C2, D6.24(C4○D4), Dic3.Q8⋊34C2, (C2×C6).237C24, (C2×C12).89C23, D6.D4.2C2, Dic6⋊C4⋊36C2, Dic3⋊5D4.12C2, (C4×C12).240C22, D6⋊C4.137C22, Dic3.30(C4○D4), (C2×D12).165C22, Dic3⋊C4.53C22, C4⋊Dic3.242C22, C22.258(S3×C23), (C22×S3).222C23, C2.30(Q8.15D6), C3⋊9(C22.46C24), (C2×Dic3).259C23, (C2×Dic6).181C22, (C4×Dic3).215C22, (S3×C4⋊C4)⋊37C2, C4⋊C4⋊7S3⋊36C2, C4⋊C4⋊S3⋊35C2, C2.88(S3×C4○D4), C6.199(C2×C4○D4), (S3×C2×C4).127C22, (C3×C42.C2)⋊10C2, (C3×C4⋊C4).192C22, (C2×C4).204(C22×S3), SmallGroup(192,1252)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.151D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c5 >
Subgroups: 496 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22.46C24, C42⋊2S3, Dic6⋊C4, Dic3.Q8, S3×C4⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, D6⋊Q8, C4⋊C4⋊S3, C3×C42.C2, C42.151D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, S3×C23, C22.46C24, Q8.15D6, S3×C4○D4, C42.151D6
►On 96 points
(1 81 7 75)(2 58 8 52)(3 83 9 77)(4 60 10 54)(5 73 11 79)(6 50 12 56)(13 91 19 85)(14 61 20 67)(15 93 21 87)(16 63 22 69)(17 95 23 89)(18 65 24 71)(25 78 31 84)(26 55 32 49)(27 80 33 74)(28 57 34 51)(29 82 35 76)(30 59 36 53)(37 90 43 96)(38 72 44 66)(39 92 45 86)(40 62 46 68)(41 94 47 88)(42 64 48 70)
(1 71 28 90)(2 66 29 85)(3 61 30 92)(4 68 31 87)(5 63 32 94)(6 70 33 89)(7 65 34 96)(8 72 35 91)(9 67 36 86)(10 62 25 93)(11 69 26 88)(12 64 27 95)(13 58 38 82)(14 53 39 77)(15 60 40 84)(16 55 41 79)(17 50 42 74)(18 57 43 81)(19 52 44 76)(20 59 45 83)(21 54 46 78)(22 49 47 73)(23 56 48 80)(24 51 37 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 14 7 20)(2 19 8 13)(3 24 9 18)(4 17 10 23)(5 22 11 16)(6 15 12 21)(25 48 31 42)(26 41 32 47)(27 46 33 40)(28 39 34 45)(29 44 35 38)(30 37 36 43)(49 69 55 63)(50 62 56 68)(51 67 57 61)(52 72 58 66)(53 65 59 71)(54 70 60 64)(73 88 79 94)(74 93 80 87)(75 86 81 92)(76 91 82 85)(77 96 83 90)(78 89 84 95)
G:=sub<Sym(96)| (1,81,7,75)(2,58,8,52)(3,83,9,77)(4,60,10,54)(5,73,11,79)(6,50,12,56)(13,91,19,85)(14,61,20,67)(15,93,21,87)(16,63,22,69)(17,95,23,89)(18,65,24,71)(25,78,31,84)(26,55,32,49)(27,80,33,74)(28,57,34,51)(29,82,35,76)(30,59,36,53)(37,90,43,96)(38,72,44,66)(39,92,45,86)(40,62,46,68)(41,94,47,88)(42,64,48,70), (1,71,28,90)(2,66,29,85)(3,61,30,92)(4,68,31,87)(5,63,32,94)(6,70,33,89)(7,65,34,96)(8,72,35,91)(9,67,36,86)(10,62,25,93)(11,69,26,88)(12,64,27,95)(13,58,38,82)(14,53,39,77)(15,60,40,84)(16,55,41,79)(17,50,42,74)(18,57,43,81)(19,52,44,76)(20,59,45,83)(21,54,46,78)(22,49,47,73)(23,56,48,80)(24,51,37,75), (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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,48,31,42)(26,41,32,47)(27,46,33,40)(28,39,34,45)(29,44,35,38)(30,37,36,43)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64)(73,88,79,94)(74,93,80,87)(75,86,81,92)(76,91,82,85)(77,96,83,90)(78,89,84,95)>;
G:=Group( (1,81,7,75)(2,58,8,52)(3,83,9,77)(4,60,10,54)(5,73,11,79)(6,50,12,56)(13,91,19,85)(14,61,20,67)(15,93,21,87)(16,63,22,69)(17,95,23,89)(18,65,24,71)(25,78,31,84)(26,55,32,49)(27,80,33,74)(28,57,34,51)(29,82,35,76)(30,59,36,53)(37,90,43,96)(38,72,44,66)(39,92,45,86)(40,62,46,68)(41,94,47,88)(42,64,48,70), (1,71,28,90)(2,66,29,85)(3,61,30,92)(4,68,31,87)(5,63,32,94)(6,70,33,89)(7,65,34,96)(8,72,35,91)(9,67,36,86)(10,62,25,93)(11,69,26,88)(12,64,27,95)(13,58,38,82)(14,53,39,77)(15,60,40,84)(16,55,41,79)(17,50,42,74)(18,57,43,81)(19,52,44,76)(20,59,45,83)(21,54,46,78)(22,49,47,73)(23,56,48,80)(24,51,37,75), (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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,48,31,42)(26,41,32,47)(27,46,33,40)(28,39,34,45)(29,44,35,38)(30,37,36,43)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64)(73,88,79,94)(74,93,80,87)(75,86,81,92)(76,91,82,85)(77,96,83,90)(78,89,84,95) );
G=PermutationGroup([[(1,81,7,75),(2,58,8,52),(3,83,9,77),(4,60,10,54),(5,73,11,79),(6,50,12,56),(13,91,19,85),(14,61,20,67),(15,93,21,87),(16,63,22,69),(17,95,23,89),(18,65,24,71),(25,78,31,84),(26,55,32,49),(27,80,33,74),(28,57,34,51),(29,82,35,76),(30,59,36,53),(37,90,43,96),(38,72,44,66),(39,92,45,86),(40,62,46,68),(41,94,47,88),(42,64,48,70)], [(1,71,28,90),(2,66,29,85),(3,61,30,92),(4,68,31,87),(5,63,32,94),(6,70,33,89),(7,65,34,96),(8,72,35,91),(9,67,36,86),(10,62,25,93),(11,69,26,88),(12,64,27,95),(13,58,38,82),(14,53,39,77),(15,60,40,84),(16,55,41,79),(17,50,42,74),(18,57,43,81),(19,52,44,76),(20,59,45,83),(21,54,46,78),(22,49,47,73),(23,56,48,80),(24,51,37,75)], [(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)], [(1,14,7,20),(2,19,8,13),(3,24,9,18),(4,17,10,23),(5,22,11,16),(6,15,12,21),(25,48,31,42),(26,41,32,47),(27,46,33,40),(28,39,34,45),(29,44,35,38),(30,37,36,43),(49,69,55,63),(50,62,56,68),(51,67,57,61),(52,72,58,66),(53,65,59,71),(54,70,60,64),(73,88,79,94),(74,93,80,87),(75,86,81,92),(76,91,82,85),(77,96,83,90),(78,89,84,95)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | 2- 1+4 | Q8.15D6 | S3×C4○D4 |
| kernel | C42.151D6 | C42⋊2S3 | Dic6⋊C4 | Dic3.Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 1 | 2 | 4 |
Matrix representation of C42.151D6 ►in GL6(𝔽13)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
C42.151D6 in GAP, Magma, Sage, TeX
C_4^2._{151}D_6 % in TeX
G:=Group("C4^2.151D6"); // GroupNames label
G:=SmallGroup(192,1252);
// by ID
G=gap.SmallGroup(192,1252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,346,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations