metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.132D6, C6.122- 1+4, (C4×Q8)⋊18S3, C4⋊C4.299D6, (Q8×C12)⋊16C2, D6⋊3Q8⋊9C2, (C4×D12).22C2, Dic3.Q8⋊9C2, (C2×Q8).204D6, C42⋊3S3⋊18C2, C42⋊2S3⋊34C2, D6.18(C4○D4), C4.68(C4○D12), (C2×C6).125C24, D6⋊C4.89C22, C12.6Q8⋊19C2, D6.D4.1C2, C12.119(C4○D4), (C4×C12).177C22, (C2×C12).623C23, (C6×Q8).225C22, (C2×D12).218C22, Dic3⋊C4.76C22, (C22×S3).47C23, C4⋊Dic3.309C22, C22.146(S3×C23), (C2×Dic3).56C23, C2.13(Q8.15D6), C3⋊5(C22.46C24), (C4×Dic3).209C22, (S3×C4⋊C4)⋊19C2, C4⋊C4⋊7S3⋊17C2, C2.32(S3×C4○D4), C2.64(C2×C4○D12), C6.147(C2×C4○D4), (S3×C2×C4).75C22, (C3×C4⋊C4).353C22, (C2×C4).289(C22×S3), SmallGroup(192,1140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.132D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 488 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C6×Q8, C22.46C24, C12.6Q8, C42⋊2S3, C4×D12, C42⋊3S3, Dic3.Q8, S3×C4⋊C4, C4⋊C4⋊7S3, D6.D4, D6⋊3Q8, Q8×C12, C42.132D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.46C24, C2×C4○D12, Q8.15D6, S3×C4○D4, C42.132D6
►On 96 points
(1 73 61 52)(2 53 62 74)(3 75 63 54)(4 55 64 76)(5 77 65 56)(6 57 66 78)(7 79 67 58)(8 59 68 80)(9 81 69 60)(10 49 70 82)(11 83 71 50)(12 51 72 84)(13 85 40 26)(14 27 41 86)(15 87 42 28)(16 29 43 88)(17 89 44 30)(18 31 45 90)(19 91 46 32)(20 33 47 92)(21 93 48 34)(22 35 37 94)(23 95 38 36)(24 25 39 96)
(1 46 67 13)(2 47 68 14)(3 48 69 15)(4 37 70 16)(5 38 71 17)(6 39 72 18)(7 40 61 19)(8 41 62 20)(9 42 63 21)(10 43 64 22)(11 44 65 23)(12 45 66 24)(25 51 90 78)(26 52 91 79)(27 53 92 80)(28 54 93 81)(29 55 94 82)(30 56 95 83)(31 57 96 84)(32 58 85 73)(33 59 86 74)(34 60 87 75)(35 49 88 76)(36 50 89 77)
(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 78 19 84)(14 83 20 77)(15 76 21 82)(16 81 22 75)(17 74 23 80)(18 79 24 73)(37 54 43 60)(38 59 44 53)(39 52 45 58)(40 57 46 51)(41 50 47 56)(42 55 48 49)(61 90 67 96)(62 95 68 89)(63 88 69 94)(64 93 70 87)(65 86 71 92)(66 91 72 85)
G:=sub<Sym(96)| (1,73,61,52)(2,53,62,74)(3,75,63,54)(4,55,64,76)(5,77,65,56)(6,57,66,78)(7,79,67,58)(8,59,68,80)(9,81,69,60)(10,49,70,82)(11,83,71,50)(12,51,72,84)(13,85,40,26)(14,27,41,86)(15,87,42,28)(16,29,43,88)(17,89,44,30)(18,31,45,90)(19,91,46,32)(20,33,47,92)(21,93,48,34)(22,35,37,94)(23,95,38,36)(24,25,39,96), (1,46,67,13)(2,47,68,14)(3,48,69,15)(4,37,70,16)(5,38,71,17)(6,39,72,18)(7,40,61,19)(8,41,62,20)(9,42,63,21)(10,43,64,22)(11,44,65,23)(12,45,66,24)(25,51,90,78)(26,52,91,79)(27,53,92,80)(28,54,93,81)(29,55,94,82)(30,56,95,83)(31,57,96,84)(32,58,85,73)(33,59,86,74)(34,60,87,75)(35,49,88,76)(36,50,89,77), (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)(13,78,19,84)(14,83,20,77)(15,76,21,82)(16,81,22,75)(17,74,23,80)(18,79,24,73)(37,54,43,60)(38,59,44,53)(39,52,45,58)(40,57,46,51)(41,50,47,56)(42,55,48,49)(61,90,67,96)(62,95,68,89)(63,88,69,94)(64,93,70,87)(65,86,71,92)(66,91,72,85)>;
G:=Group( (1,73,61,52)(2,53,62,74)(3,75,63,54)(4,55,64,76)(5,77,65,56)(6,57,66,78)(7,79,67,58)(8,59,68,80)(9,81,69,60)(10,49,70,82)(11,83,71,50)(12,51,72,84)(13,85,40,26)(14,27,41,86)(15,87,42,28)(16,29,43,88)(17,89,44,30)(18,31,45,90)(19,91,46,32)(20,33,47,92)(21,93,48,34)(22,35,37,94)(23,95,38,36)(24,25,39,96), (1,46,67,13)(2,47,68,14)(3,48,69,15)(4,37,70,16)(5,38,71,17)(6,39,72,18)(7,40,61,19)(8,41,62,20)(9,42,63,21)(10,43,64,22)(11,44,65,23)(12,45,66,24)(25,51,90,78)(26,52,91,79)(27,53,92,80)(28,54,93,81)(29,55,94,82)(30,56,95,83)(31,57,96,84)(32,58,85,73)(33,59,86,74)(34,60,87,75)(35,49,88,76)(36,50,89,77), (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)(13,78,19,84)(14,83,20,77)(15,76,21,82)(16,81,22,75)(17,74,23,80)(18,79,24,73)(37,54,43,60)(38,59,44,53)(39,52,45,58)(40,57,46,51)(41,50,47,56)(42,55,48,49)(61,90,67,96)(62,95,68,89)(63,88,69,94)(64,93,70,87)(65,86,71,92)(66,91,72,85) );
G=PermutationGroup([[(1,73,61,52),(2,53,62,74),(3,75,63,54),(4,55,64,76),(5,77,65,56),(6,57,66,78),(7,79,67,58),(8,59,68,80),(9,81,69,60),(10,49,70,82),(11,83,71,50),(12,51,72,84),(13,85,40,26),(14,27,41,86),(15,87,42,28),(16,29,43,88),(17,89,44,30),(18,31,45,90),(19,91,46,32),(20,33,47,92),(21,93,48,34),(22,35,37,94),(23,95,38,36),(24,25,39,96)], [(1,46,67,13),(2,47,68,14),(3,48,69,15),(4,37,70,16),(5,38,71,17),(6,39,72,18),(7,40,61,19),(8,41,62,20),(9,42,63,21),(10,43,64,22),(11,44,65,23),(12,45,66,24),(25,51,90,78),(26,52,91,79),(27,53,92,80),(28,54,93,81),(29,55,94,82),(30,56,95,83),(31,57,96,84),(32,58,85,73),(33,59,86,74),(34,60,87,75),(35,49,88,76),(36,50,89,77)], [(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),(13,78,19,84),(14,83,20,77),(15,76,21,82),(16,81,22,75),(17,74,23,80),(18,79,24,73),(37,54,43,60),(38,59,44,53),(39,52,45,58),(40,57,46,51),(41,50,47,56),(42,55,48,49),(61,90,67,96),(62,95,68,89),(63,88,69,94),(64,93,70,87),(65,86,71,92),(66,91,72,85)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 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 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- 1+4 | Q8.15D6 | S3×C4○D4 |
| kernel | C42.132D6 | C12.6Q8 | C42⋊2S3 | C4×D12 | C42⋊3S3 | Dic3.Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6.D4 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.132D6 ►in GL4(𝔽13) generated by
| 5 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 6 | 3 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 2 | 11 |
| 0 | 0 | 9 | 11 |
G:=sub<GL(4,GF(13))| [5,0,0,0,8,8,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,10,6,0,0,7,3],[1,2,0,0,0,12,0,0,0,0,8,5,0,0,8,0],[1,2,0,0,0,12,0,0,0,0,2,9,0,0,11,11] >;
C42.132D6 in GAP, Magma, Sage, TeX
C_4^2._{132}D_6 % in TeX
G:=Group("C4^2.132D6"); // GroupNames label
G:=SmallGroup(192,1140);
// by ID
G=gap.SmallGroup(192,1140);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,136,6278]);
// 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*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations