metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.150D6, C6.282- 1+4, C6.1322+ 1+4, (C4×D12)⋊47C2, C4⋊C4.207D6, C42.C2⋊6S3, C42⋊3S3⋊9C2, C4.D12⋊37C2, D6⋊Q8⋊35C2, Dic3⋊5D4⋊36C2, D6.11(C4○D4), C12⋊D4.12C2, D6.D4⋊34C2, C2.57(D4○D12), Dic3.Q8⋊33C2, (C2×C6).236C24, D6⋊C4.10C22, (C4×C12).196C22, (C2×C12).188C23, (C2×D12).164C22, Dic3⋊C4.52C22, C4⋊Dic3.314C22, C22.257(S3×C23), (C22×S3).102C23, C2.29(Q8.15D6), C3⋊8(C22.33C24), (C2×Dic3).258C23, (C2×Dic6).180C22, (C4×Dic3).143C22, (S3×C4⋊C4)⋊36C2, C4⋊C4⋊S3⋊34C2, C2.87(S3×C4○D4), C6.198(C2×C4○D4), (C3×C42.C2)⋊9C2, (S3×C2×C4).126C22, (C2×C4).80(C22×S3), (C3×C4⋊C4).191C22, SmallGroup(192,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.150D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 576 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, Q8, C23, 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, C4×D4, C4⋊D4, 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, C22.33C24, C4×D12, C42⋊3S3, Dic3.Q8, S3×C4⋊C4, Dic3⋊5D4, D6.D4, C12⋊D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.150D6
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, Q8.15D6, S3×C4○D4, D4○D12, C42.150D6
►On 96 points
(1 19 46 49)(2 14 47 56)(3 21 48 51)(4 16 37 58)(5 23 38 53)(6 18 39 60)(7 13 40 55)(8 20 41 50)(9 15 42 57)(10 22 43 52)(11 17 44 59)(12 24 45 54)(25 73 67 85)(26 80 68 92)(27 75 69 87)(28 82 70 94)(29 77 71 89)(30 84 72 96)(31 79 61 91)(32 74 62 86)(33 81 63 93)(34 76 64 88)(35 83 65 95)(36 78 66 90)
(1 86 40 80)(2 81 41 87)(3 88 42 82)(4 83 43 89)(5 90 44 84)(6 73 45 91)(7 92 46 74)(8 75 47 93)(9 94 48 76)(10 77 37 95)(11 96 38 78)(12 79 39 85)(13 26 49 62)(14 63 50 27)(15 28 51 64)(16 65 52 29)(17 30 53 66)(18 67 54 31)(19 32 55 68)(20 69 56 33)(21 34 57 70)(22 71 58 35)(23 36 59 72)(24 61 60 25)
(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 31 7 25)(2 36 8 30)(3 29 9 35)(4 34 10 28)(5 27 11 33)(6 32 12 26)(13 79 19 73)(14 84 20 78)(15 77 21 83)(16 82 22 76)(17 75 23 81)(18 80 24 74)(37 64 43 70)(38 69 44 63)(39 62 45 68)(40 67 46 61)(41 72 47 66)(42 65 48 71)(49 85 55 91)(50 90 56 96)(51 95 57 89)(52 88 58 94)(53 93 59 87)(54 86 60 92)
G:=sub<Sym(96)| (1,19,46,49)(2,14,47,56)(3,21,48,51)(4,16,37,58)(5,23,38,53)(6,18,39,60)(7,13,40,55)(8,20,41,50)(9,15,42,57)(10,22,43,52)(11,17,44,59)(12,24,45,54)(25,73,67,85)(26,80,68,92)(27,75,69,87)(28,82,70,94)(29,77,71,89)(30,84,72,96)(31,79,61,91)(32,74,62,86)(33,81,63,93)(34,76,64,88)(35,83,65,95)(36,78,66,90), (1,86,40,80)(2,81,41,87)(3,88,42,82)(4,83,43,89)(5,90,44,84)(6,73,45,91)(7,92,46,74)(8,75,47,93)(9,94,48,76)(10,77,37,95)(11,96,38,78)(12,79,39,85)(13,26,49,62)(14,63,50,27)(15,28,51,64)(16,65,52,29)(17,30,53,66)(18,67,54,31)(19,32,55,68)(20,69,56,33)(21,34,57,70)(22,71,58,35)(23,36,59,72)(24,61,60,25), (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), (1,31,7,25)(2,36,8,30)(3,29,9,35)(4,34,10,28)(5,27,11,33)(6,32,12,26)(13,79,19,73)(14,84,20,78)(15,77,21,83)(16,82,22,76)(17,75,23,81)(18,80,24,74)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(49,85,55,91)(50,90,56,96)(51,95,57,89)(52,88,58,94)(53,93,59,87)(54,86,60,92)>;
G:=Group( (1,19,46,49)(2,14,47,56)(3,21,48,51)(4,16,37,58)(5,23,38,53)(6,18,39,60)(7,13,40,55)(8,20,41,50)(9,15,42,57)(10,22,43,52)(11,17,44,59)(12,24,45,54)(25,73,67,85)(26,80,68,92)(27,75,69,87)(28,82,70,94)(29,77,71,89)(30,84,72,96)(31,79,61,91)(32,74,62,86)(33,81,63,93)(34,76,64,88)(35,83,65,95)(36,78,66,90), (1,86,40,80)(2,81,41,87)(3,88,42,82)(4,83,43,89)(5,90,44,84)(6,73,45,91)(7,92,46,74)(8,75,47,93)(9,94,48,76)(10,77,37,95)(11,96,38,78)(12,79,39,85)(13,26,49,62)(14,63,50,27)(15,28,51,64)(16,65,52,29)(17,30,53,66)(18,67,54,31)(19,32,55,68)(20,69,56,33)(21,34,57,70)(22,71,58,35)(23,36,59,72)(24,61,60,25), (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), (1,31,7,25)(2,36,8,30)(3,29,9,35)(4,34,10,28)(5,27,11,33)(6,32,12,26)(13,79,19,73)(14,84,20,78)(15,77,21,83)(16,82,22,76)(17,75,23,81)(18,80,24,74)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(49,85,55,91)(50,90,56,96)(51,95,57,89)(52,88,58,94)(53,93,59,87)(54,86,60,92) );
G=PermutationGroup([[(1,19,46,49),(2,14,47,56),(3,21,48,51),(4,16,37,58),(5,23,38,53),(6,18,39,60),(7,13,40,55),(8,20,41,50),(9,15,42,57),(10,22,43,52),(11,17,44,59),(12,24,45,54),(25,73,67,85),(26,80,68,92),(27,75,69,87),(28,82,70,94),(29,77,71,89),(30,84,72,96),(31,79,61,91),(32,74,62,86),(33,81,63,93),(34,76,64,88),(35,83,65,95),(36,78,66,90)], [(1,86,40,80),(2,81,41,87),(3,88,42,82),(4,83,43,89),(5,90,44,84),(6,73,45,91),(7,92,46,74),(8,75,47,93),(9,94,48,76),(10,77,37,95),(11,96,38,78),(12,79,39,85),(13,26,49,62),(14,63,50,27),(15,28,51,64),(16,65,52,29),(17,30,53,66),(18,67,54,31),(19,32,55,68),(20,69,56,33),(21,34,57,70),(22,71,58,35),(23,36,59,72),(24,61,60,25)], [(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)], [(1,31,7,25),(2,36,8,30),(3,29,9,35),(4,34,10,28),(5,27,11,33),(6,32,12,26),(13,79,19,73),(14,84,20,78),(15,77,21,83),(16,82,22,76),(17,75,23,81),(18,80,24,74),(37,64,43,70),(38,69,44,63),(39,62,45,68),(40,67,46,61),(41,72,47,66),(42,65,48,71),(49,85,55,91),(50,90,56,96),(51,95,57,89),(52,88,58,94),(53,93,59,87),(54,86,60,92)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | Q8.15D6 | S3×C4○D4 | D4○D12 |
| kernel | C42.150D6 | C4×D12 | C42⋊3S3 | Dic3.Q8 | S3×C4⋊C4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | D6 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 6 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.150D6 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 7 | 6 |
| 0 | 0 | 8 | 3 | 7 | 1 |
| 0 | 0 | 7 | 6 | 5 | 8 |
| 0 | 0 | 7 | 1 | 5 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 9 | 9 |
| 0 | 0 | 0 | 3 | 0 | 4 |
| 0 | 0 | 9 | 9 | 3 | 3 |
| 0 | 0 | 0 | 4 | 0 | 10 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,8,7,7,0,0,5,3,6,1,0,0,7,7,5,5,0,0,6,1,8,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,0,9,0,0,0,10,3,9,4,0,0,9,0,3,0,0,0,9,4,3,10] >;
C42.150D6 in GAP, Magma, Sage, TeX
C_4^2._{150}D_6 % in TeX
G:=Group("C4^2.150D6"); // GroupNames label
G:=SmallGroup(192,1251);
// by ID
G=gap.SmallGroup(192,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,570,136,6278]);
// 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=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations