metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.153D6, C6.1332+ 1+4, (C4×D12)⋊48C2, C4⋊C4.209D6, C42.C2⋊9S3, C12⋊D4⋊33C2, Dic3⋊5D4⋊37C2, D6.32(C4○D4), D6.D4⋊35C2, C2.58(D4○D12), (C2×C12).91C23, (C2×C6).239C24, D6⋊C4.41C22, C12.130(C4○D4), (C4×C12).198C22, C4.39(Q8⋊3S3), (C2×D12).268C22, Dic3⋊C4.54C22, C4⋊Dic3.315C22, C22.260(S3×C23), (C22×S3).104C23, (C2×Dic3).124C23, (C4×Dic3).145C22, C3⋊10(C22.47C24), (S3×C4⋊C4)⋊39C2, C4⋊C4⋊S3⋊37C2, C4⋊C4⋊7S3⋊38C2, C2.90(S3×C4○D4), C6.201(C2×C4○D4), (S3×C2×C4).129C22, (C2×C4).82(C22×S3), C2.24(C2×Q8⋊3S3), (C3×C42.C2)⋊12C2, (C3×C4⋊C4).194C22, SmallGroup(192,1254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.153D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 656 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C22.47C24, C4×D12, S3×C4⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, C12⋊D4, C12⋊D4, C4⋊C4⋊S3, C3×C42.C2, C42.153D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, Q8⋊3S3, S3×C23, C22.47C24, C2×Q8⋊3S3, S3×C4○D4, D4○D12, C42.153D6
►On 96 points
(1 57 66 31)(2 32 67 58)(3 59 68 33)(4 34 69 60)(5 49 70 35)(6 36 71 50)(7 51 72 25)(8 26 61 52)(9 53 62 27)(10 28 63 54)(11 55 64 29)(12 30 65 56)(13 76 40 89)(14 90 41 77)(15 78 42 91)(16 92 43 79)(17 80 44 93)(18 94 45 81)(19 82 46 95)(20 96 47 83)(21 84 48 85)(22 86 37 73)(23 74 38 87)(24 88 39 75)
(1 83 72 90)(2 78 61 85)(3 73 62 92)(4 80 63 87)(5 75 64 94)(6 82 65 89)(7 77 66 96)(8 84 67 91)(9 79 68 86)(10 74 69 93)(11 81 70 88)(12 76 71 95)(13 36 46 56)(14 31 47 51)(15 26 48 58)(16 33 37 53)(17 28 38 60)(18 35 39 55)(19 30 40 50)(20 25 41 57)(21 32 42 52)(22 27 43 59)(23 34 44 54)(24 29 45 49)
(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 37 7 43)(2 42 8 48)(3 47 9 41)(4 40 10 46)(5 45 11 39)(6 38 12 44)(13 63 19 69)(14 68 20 62)(15 61 21 67)(16 66 22 72)(17 71 23 65)(18 64 24 70)(25 79 31 73)(26 84 32 78)(27 77 33 83)(28 82 34 76)(29 75 35 81)(30 80 36 74)(49 94 55 88)(50 87 56 93)(51 92 57 86)(52 85 58 91)(53 90 59 96)(54 95 60 89)
G:=sub<Sym(96)| (1,57,66,31)(2,32,67,58)(3,59,68,33)(4,34,69,60)(5,49,70,35)(6,36,71,50)(7,51,72,25)(8,26,61,52)(9,53,62,27)(10,28,63,54)(11,55,64,29)(12,30,65,56)(13,76,40,89)(14,90,41,77)(15,78,42,91)(16,92,43,79)(17,80,44,93)(18,94,45,81)(19,82,46,95)(20,96,47,83)(21,84,48,85)(22,86,37,73)(23,74,38,87)(24,88,39,75), (1,83,72,90)(2,78,61,85)(3,73,62,92)(4,80,63,87)(5,75,64,94)(6,82,65,89)(7,77,66,96)(8,84,67,91)(9,79,68,86)(10,74,69,93)(11,81,70,88)(12,76,71,95)(13,36,46,56)(14,31,47,51)(15,26,48,58)(16,33,37,53)(17,28,38,60)(18,35,39,55)(19,30,40,50)(20,25,41,57)(21,32,42,52)(22,27,43,59)(23,34,44,54)(24,29,45,49), (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), (1,37,7,43)(2,42,8,48)(3,47,9,41)(4,40,10,46)(5,45,11,39)(6,38,12,44)(13,63,19,69)(14,68,20,62)(15,61,21,67)(16,66,22,72)(17,71,23,65)(18,64,24,70)(25,79,31,73)(26,84,32,78)(27,77,33,83)(28,82,34,76)(29,75,35,81)(30,80,36,74)(49,94,55,88)(50,87,56,93)(51,92,57,86)(52,85,58,91)(53,90,59,96)(54,95,60,89)>;
G:=Group( (1,57,66,31)(2,32,67,58)(3,59,68,33)(4,34,69,60)(5,49,70,35)(6,36,71,50)(7,51,72,25)(8,26,61,52)(9,53,62,27)(10,28,63,54)(11,55,64,29)(12,30,65,56)(13,76,40,89)(14,90,41,77)(15,78,42,91)(16,92,43,79)(17,80,44,93)(18,94,45,81)(19,82,46,95)(20,96,47,83)(21,84,48,85)(22,86,37,73)(23,74,38,87)(24,88,39,75), (1,83,72,90)(2,78,61,85)(3,73,62,92)(4,80,63,87)(5,75,64,94)(6,82,65,89)(7,77,66,96)(8,84,67,91)(9,79,68,86)(10,74,69,93)(11,81,70,88)(12,76,71,95)(13,36,46,56)(14,31,47,51)(15,26,48,58)(16,33,37,53)(17,28,38,60)(18,35,39,55)(19,30,40,50)(20,25,41,57)(21,32,42,52)(22,27,43,59)(23,34,44,54)(24,29,45,49), (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), (1,37,7,43)(2,42,8,48)(3,47,9,41)(4,40,10,46)(5,45,11,39)(6,38,12,44)(13,63,19,69)(14,68,20,62)(15,61,21,67)(16,66,22,72)(17,71,23,65)(18,64,24,70)(25,79,31,73)(26,84,32,78)(27,77,33,83)(28,82,34,76)(29,75,35,81)(30,80,36,74)(49,94,55,88)(50,87,56,93)(51,92,57,86)(52,85,58,91)(53,90,59,96)(54,95,60,89) );
G=PermutationGroup([[(1,57,66,31),(2,32,67,58),(3,59,68,33),(4,34,69,60),(5,49,70,35),(6,36,71,50),(7,51,72,25),(8,26,61,52),(9,53,62,27),(10,28,63,54),(11,55,64,29),(12,30,65,56),(13,76,40,89),(14,90,41,77),(15,78,42,91),(16,92,43,79),(17,80,44,93),(18,94,45,81),(19,82,46,95),(20,96,47,83),(21,84,48,85),(22,86,37,73),(23,74,38,87),(24,88,39,75)], [(1,83,72,90),(2,78,61,85),(3,73,62,92),(4,80,63,87),(5,75,64,94),(6,82,65,89),(7,77,66,96),(8,84,67,91),(9,79,68,86),(10,74,69,93),(11,81,70,88),(12,76,71,95),(13,36,46,56),(14,31,47,51),(15,26,48,58),(16,33,37,53),(17,28,38,60),(18,35,39,55),(19,30,40,50),(20,25,41,57),(21,32,42,52),(22,27,43,59),(23,34,44,54),(24,29,45,49)], [(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)], [(1,37,7,43),(2,42,8,48),(3,47,9,41),(4,40,10,46),(5,45,11,39),(6,38,12,44),(13,63,19,69),(14,68,20,62),(15,61,21,67),(16,66,22,72),(17,71,23,65),(18,64,24,70),(25,79,31,73),(26,84,32,78),(27,77,33,83),(28,82,34,76),(29,75,35,81),(30,80,36,74),(49,94,55,88),(50,87,56,93),(51,92,57,86),(52,85,58,91),(53,90,59,96),(54,95,60,89)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | 2+ 1+4 | Q8⋊3S3 | S3×C4○D4 | D4○D12 |
| kernel | C42.153D6 | C4×D12 | S3×C4⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C42.153D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,11,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,3,0,0,0,0,8,1],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C42.153D6 in GAP, Magma, Sage, TeX
C_4^2._{153}D_6 % in TeX
G:=Group("C4^2.153D6"); // GroupNames label
G:=SmallGroup(192,1254);
// by ID
G=gap.SmallGroup(192,1254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,185,192,6278]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations