metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.31D6, C4⋊C8⋊16S3, D6⋊C4.15C4, D6⋊C8.12C2, C12⋊C8⋊16C2, C24⋊C4⋊22C2, (C2×C8).218D6, (C8×Dic3)⋊24C2, C6.14(C8○D4), C2.16(C8○D12), Dic3⋊C4.15C4, C42⋊2S3.2C2, (C4×C12).66C22, C12.337(C4○D4), C2.15(D12.C4), (C2×C12).837C23, (C2×C24).259C22, C4.57(Q8⋊3S3), C4.132(D4⋊2S3), C6.33(C42⋊C2), C3⋊4(C42.7C22), (C4×Dic3).278C22, (C3×C4⋊C8)⋊26C2, (C2×C4).37(C4×S3), C22.115(S3×C2×C4), (C2×C12).182(C2×C4), (C2×C3⋊C8).197C22, (S3×C2×C4).182C22, (C2×C6).92(C22×C4), C2.10(C4⋊C4⋊7S3), (C22×S3).17(C2×C4), (C2×C4).779(C22×S3), (C2×Dic3).55(C2×C4), SmallGroup(192,399)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.31D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b2c5 >
Subgroups: 216 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C42⋊C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C42.7C22, C12⋊C8, C8×Dic3, C24⋊C4, D6⋊C8, C3×C4⋊C8, C42⋊2S3, C42.31D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C8○D4, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C42.7C22, C4⋊C4⋊7S3, C8○D12, D12.C4, C42.31D6
►On 96 points
Generators in S96
(1 50 34 79)(2 80 35 51)(3 52 36 81)(4 82 37 53)(5 54 38 83)(6 84 39 55)(7 56 40 85)(8 86 41 57)(9 58 42 87)(10 88 43 59)(11 60 44 89)(12 90 45 61)(13 62 46 91)(14 92 47 63)(15 64 48 93)(16 94 25 65)(17 66 26 95)(18 96 27 67)(19 68 28 73)(20 74 29 69)(21 70 30 75)(22 76 31 71)(23 72 32 77)(24 78 33 49)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)
(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 18 28 45 13 6 40 33)(2 44 29 5 14 32 41 17)(3 4 30 31 15 16 42 43)(7 24 34 27 19 12 46 39)(8 26 35 11 20 38 47 23)(9 10 36 37 21 22 48 25)(49 91 96 56 61 79 84 68)(50 55 73 78 62 67 85 90)(51 77 74 66 63 89 86 54)(52 65 75 88 64 53 87 76)(57 83 80 72 69 95 92 60)(58 71 81 94 70 59 93 82)
G:=sub<Sym(96)| (1,50,34,79)(2,80,35,51)(3,52,36,81)(4,82,37,53)(5,54,38,83)(6,84,39,55)(7,56,40,85)(8,86,41,57)(9,58,42,87)(10,88,43,59)(11,60,44,89)(12,90,45,61)(13,62,46,91)(14,92,47,63)(15,64,48,93)(16,94,25,65)(17,66,26,95)(18,96,27,67)(19,68,28,73)(20,74,29,69)(21,70,30,75)(22,76,31,71)(23,72,32,77)(24,78,33,49), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,28,45,13,6,40,33)(2,44,29,5,14,32,41,17)(3,4,30,31,15,16,42,43)(7,24,34,27,19,12,46,39)(8,26,35,11,20,38,47,23)(9,10,36,37,21,22,48,25)(49,91,96,56,61,79,84,68)(50,55,73,78,62,67,85,90)(51,77,74,66,63,89,86,54)(52,65,75,88,64,53,87,76)(57,83,80,72,69,95,92,60)(58,71,81,94,70,59,93,82)>;
G:=Group( (1,50,34,79)(2,80,35,51)(3,52,36,81)(4,82,37,53)(5,54,38,83)(6,84,39,55)(7,56,40,85)(8,86,41,57)(9,58,42,87)(10,88,43,59)(11,60,44,89)(12,90,45,61)(13,62,46,91)(14,92,47,63)(15,64,48,93)(16,94,25,65)(17,66,26,95)(18,96,27,67)(19,68,28,73)(20,74,29,69)(21,70,30,75)(22,76,31,71)(23,72,32,77)(24,78,33,49), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,28,45,13,6,40,33)(2,44,29,5,14,32,41,17)(3,4,30,31,15,16,42,43)(7,24,34,27,19,12,46,39)(8,26,35,11,20,38,47,23)(9,10,36,37,21,22,48,25)(49,91,96,56,61,79,84,68)(50,55,73,78,62,67,85,90)(51,77,74,66,63,89,86,54)(52,65,75,88,64,53,87,76)(57,83,80,72,69,95,92,60)(58,71,81,94,70,59,93,82) );
G=PermutationGroup([[(1,50,34,79),(2,80,35,51),(3,52,36,81),(4,82,37,53),(5,54,38,83),(6,84,39,55),(7,56,40,85),(8,86,41,57),(9,58,42,87),(10,88,43,59),(11,60,44,89),(12,90,45,61),(13,62,46,91),(14,92,47,63),(15,64,48,93),(16,94,25,65),(17,66,26,95),(18,96,27,67),(19,68,28,73),(20,74,29,69),(21,70,30,75),(22,76,31,71),(23,72,32,77),(24,78,33,49)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84)], [(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)], [(1,18,28,45,13,6,40,33),(2,44,29,5,14,32,41,17),(3,4,30,31,15,16,42,43),(7,24,34,27,19,12,46,39),(8,26,35,11,20,38,47,23),(9,10,36,37,21,22,48,25),(49,91,96,56,61,79,84,68),(50,55,73,78,62,67,85,90),(51,77,74,66,63,89,86,54),(52,65,75,88,64,53,87,76),(57,83,80,72,69,95,92,60),(58,71,81,94,70,59,93,82)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C4 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C8○D4 | C8○D12 | D4⋊2S3 | Q8⋊3S3 | D12.C4 |
| kernel | C42.31D6 | C12⋊C8 | C8×Dic3 | C24⋊C4 | D6⋊C8 | C3×C4⋊C8 | C42⋊2S3 | Dic3⋊C4 | D6⋊C4 | C4⋊C8 | C42 | C2×C8 | C12 | C2×C4 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 1 | 1 | 2 |
Matrix representation of C42.31D6 ►in GL6(𝔽73)
| 25 | 11 | 0 | 0 | 0 | 0 |
| 3 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 46 | 46 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 0 | 0 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 26 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [25,3,0,0,0,0,11,48,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,46,0,0,0,0,0,46],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,0,0,0,0,0,0,51,0,0,0,0,0,0,0,46,0,0,0,0,27,27,0,0,0,0,0,0,72,0,0,0,0,0,71,1],[51,0,0,0,0,0,26,22,0,0,0,0,0,0,46,0,0,0,0,0,27,27,0,0,0,0,0,0,1,72,0,0,0,0,2,72] >;
C42.31D6 in GAP, Magma, Sage, TeX
C_4^2._{31}D_6 % in TeX
G:=Group("C4^2.31D6"); // GroupNames label
G:=SmallGroup(192,399);
// by ID
G=gap.SmallGroup(192,399);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,422,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations