metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.185D6, C8⋊C4⋊9S3, D6⋊C4.8C4, D6⋊C8.16C2, (C2×C8).158D6, Dic3⋊C8⋊37C2, C6.23(C8○D4), Dic3⋊C4.8C4, C2.8(D12.C4), C12.247(C4○D4), C4.131(C4○D12), (C2×C12).815C23, (C4×C12).230C22, C42⋊2S3.12C2, (C2×C24).312C22, C6.10(C42⋊C2), C2.13(C42⋊2S3), C3⋊2(C42.7C22), (C4×Dic3).181C22, (C4×C3⋊C8)⋊22C2, (C2×C4).62(C4×S3), (C3×C8⋊C4)⋊18C2, C22.100(S3×C2×C4), (C2×C12).149(C2×C4), (C2×C3⋊C8).297C22, (S3×C2×C4).178C22, (C2×C6).70(C22×C4), (C22×S3).11(C2×C4), (C2×C4).757(C22×S3), (C2×Dic3).15(C2×C4), SmallGroup(192,268)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.185D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2b2c5 >
Subgroups: 216 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, 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×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C42.7C22, C4×C3⋊C8, Dic3⋊C8, D6⋊C8, C3×C8⋊C4, C42⋊2S3, C42.185D6
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, C4○D12, C42.7C22, C42⋊2S3, D12.C4, C42.185D6
►On 96 points
(1 77 61 31)(2 90 62 44)(3 79 63 33)(4 92 64 46)(5 81 65 35)(6 94 66 48)(7 83 67 37)(8 96 68 26)(9 85 69 39)(10 74 70 28)(11 87 71 41)(12 76 72 30)(13 89 49 43)(14 78 50 32)(15 91 51 45)(16 80 52 34)(17 93 53 47)(18 82 54 36)(19 95 55 25)(20 84 56 38)(21 73 57 27)(22 86 58 40)(23 75 59 29)(24 88 60 42)
(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 55 72 13 6 67 60)(2 71 56 5 14 59 68 17)(3 4 57 58 15 16 69 70)(7 24 61 54 19 12 49 66)(8 53 62 11 20 65 50 23)(9 10 63 64 21 22 51 52)(25 30 89 94 37 42 77 82)(26 93 90 41 38 81 78 29)(27 40 91 80 39 28 79 92)(31 36 95 76 43 48 83 88)(32 75 96 47 44 87 84 35)(33 46 73 86 45 34 85 74)
G:=sub<Sym(96)| (1,77,61,31)(2,90,62,44)(3,79,63,33)(4,92,64,46)(5,81,65,35)(6,94,66,48)(7,83,67,37)(8,96,68,26)(9,85,69,39)(10,74,70,28)(11,87,71,41)(12,76,72,30)(13,89,49,43)(14,78,50,32)(15,91,51,45)(16,80,52,34)(17,93,53,47)(18,82,54,36)(19,95,55,25)(20,84,56,38)(21,73,57,27)(22,86,58,40)(23,75,59,29)(24,88,60,42), (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), (1,18,55,72,13,6,67,60)(2,71,56,5,14,59,68,17)(3,4,57,58,15,16,69,70)(7,24,61,54,19,12,49,66)(8,53,62,11,20,65,50,23)(9,10,63,64,21,22,51,52)(25,30,89,94,37,42,77,82)(26,93,90,41,38,81,78,29)(27,40,91,80,39,28,79,92)(31,36,95,76,43,48,83,88)(32,75,96,47,44,87,84,35)(33,46,73,86,45,34,85,74)>;
G:=Group( (1,77,61,31)(2,90,62,44)(3,79,63,33)(4,92,64,46)(5,81,65,35)(6,94,66,48)(7,83,67,37)(8,96,68,26)(9,85,69,39)(10,74,70,28)(11,87,71,41)(12,76,72,30)(13,89,49,43)(14,78,50,32)(15,91,51,45)(16,80,52,34)(17,93,53,47)(18,82,54,36)(19,95,55,25)(20,84,56,38)(21,73,57,27)(22,86,58,40)(23,75,59,29)(24,88,60,42), (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), (1,18,55,72,13,6,67,60)(2,71,56,5,14,59,68,17)(3,4,57,58,15,16,69,70)(7,24,61,54,19,12,49,66)(8,53,62,11,20,65,50,23)(9,10,63,64,21,22,51,52)(25,30,89,94,37,42,77,82)(26,93,90,41,38,81,78,29)(27,40,91,80,39,28,79,92)(31,36,95,76,43,48,83,88)(32,75,96,47,44,87,84,35)(33,46,73,86,45,34,85,74) );
G=PermutationGroup([[(1,77,61,31),(2,90,62,44),(3,79,63,33),(4,92,64,46),(5,81,65,35),(6,94,66,48),(7,83,67,37),(8,96,68,26),(9,85,69,39),(10,74,70,28),(11,87,71,41),(12,76,72,30),(13,89,49,43),(14,78,50,32),(15,91,51,45),(16,80,52,34),(17,93,53,47),(18,82,54,36),(19,95,55,25),(20,84,56,38),(21,73,57,27),(22,86,58,40),(23,75,59,29),(24,88,60,42)], [(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)], [(1,18,55,72,13,6,67,60),(2,71,56,5,14,59,68,17),(3,4,57,58,15,16,69,70),(7,24,61,54,19,12,49,66),(8,53,62,11,20,65,50,23),(9,10,63,64,21,22,51,52),(25,30,89,94,37,42,77,82),(26,93,90,41,38,81,78,29),(27,40,91,80,39,28,79,92),(31,36,95,76,43,48,83,88),(32,75,96,47,44,87,84,35),(33,46,73,86,45,34,85,74)]])
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 | ··· | 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C8○D4 | C4○D12 | D12.C4 |
| kernel | C42.185D6 | C4×C3⋊C8 | Dic3⋊C8 | D6⋊C8 | C3×C8⋊C4 | C42⋊2S3 | Dic3⋊C4 | D6⋊C4 | C8⋊C4 | C42 | C2×C8 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 4 |
Matrix representation of C42.185D6 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 2 | 67 |
| 0 | 0 | 13 | 71 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 43 | 30 | 0 | 0 |
| 43 | 13 | 0 | 0 |
| 0 | 0 | 20 | 37 |
| 0 | 0 | 57 | 53 |
| 30 | 43 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 20 | 13 |
| 0 | 0 | 57 | 53 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,2,13,0,0,67,71],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[43,43,0,0,30,13,0,0,0,0,20,57,0,0,37,53],[30,13,0,0,43,43,0,0,0,0,20,57,0,0,13,53] >;
C42.185D6 in GAP, Magma, Sage, TeX
C_4^2._{185}D_6 % in TeX
G:=Group("C4^2.185D6"); // GroupNames label
G:=SmallGroup(192,268);
// by ID
G=gap.SmallGroup(192,268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,422,387,58,136,6278]);
// 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*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations