metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.243D6, (C4×C8)⋊4S3, (C4×C24)⋊4C2, D6⋊C4.2C4, D6⋊C8.1C2, Dic3⋊C8⋊1C2, C6.6(C8○D4), (C2×C8).284D6, C2.8(C8○D12), Dic3⋊C4.2C4, C12.243(C4○D4), C4.127(C4○D12), (C2×C12).808C23, (C2×C24).344C22, C42.S3⋊16C2, (C4×C12).341C22, C42⋊2S3.10C2, C6.8(C42⋊C2), C2.11(C42⋊2S3), C3⋊1(C42.7C22), (C4×Dic3).179C22, (C2×C4).89(C4×S3), C22.96(S3×C2×C4), (C2×C12).206(C2×C4), (C2×C3⋊C8).188C22, (S3×C2×C4).176C22, (C2×C6).63(C22×C4), (C22×S3).10(C2×C4), (C2×C4).750(C22×S3), (C2×Dic3).12(C2×C4), SmallGroup(192,249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.243D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, ac=ca, dad-1=ab2, 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, C42.S3, Dic3⋊C8, D6⋊C8, C4×C24, C42⋊2S3, C42.243D6
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, C8○D12, C42.243D6
►On 96 points
(1 41 61 75)(2 42 62 76)(3 43 63 77)(4 44 64 78)(5 45 65 79)(6 46 66 80)(7 47 67 81)(8 48 68 82)(9 25 69 83)(10 26 70 84)(11 27 71 85)(12 28 72 86)(13 29 49 87)(14 30 50 88)(15 31 51 89)(16 32 52 90)(17 33 53 91)(18 34 54 92)(19 35 55 93)(20 36 56 94)(21 37 57 95)(22 38 58 96)(23 39 59 73)(24 40 60 74)
(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 38 77 90 37 26 89 78)(27 48 79 76 39 36 91 88)(28 75 80 35 40 87 92 47)(29 34 81 86 41 46 93 74)(30 85 82 45 42 73 94 33)(31 44 83 96 43 32 95 84)
G:=sub<Sym(96)| (1,41,61,75)(2,42,62,76)(3,43,63,77)(4,44,64,78)(5,45,65,79)(6,46,66,80)(7,47,67,81)(8,48,68,82)(9,25,69,83)(10,26,70,84)(11,27,71,85)(12,28,72,86)(13,29,49,87)(14,30,50,88)(15,31,51,89)(16,32,52,90)(17,33,53,91)(18,34,54,92)(19,35,55,93)(20,36,56,94)(21,37,57,95)(22,38,58,96)(23,39,59,73)(24,40,60,74), (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)(25,38,77,90,37,26,89,78)(27,48,79,76,39,36,91,88)(28,75,80,35,40,87,92,47)(29,34,81,86,41,46,93,74)(30,85,82,45,42,73,94,33)(31,44,83,96,43,32,95,84)>;
G:=Group( (1,41,61,75)(2,42,62,76)(3,43,63,77)(4,44,64,78)(5,45,65,79)(6,46,66,80)(7,47,67,81)(8,48,68,82)(9,25,69,83)(10,26,70,84)(11,27,71,85)(12,28,72,86)(13,29,49,87)(14,30,50,88)(15,31,51,89)(16,32,52,90)(17,33,53,91)(18,34,54,92)(19,35,55,93)(20,36,56,94)(21,37,57,95)(22,38,58,96)(23,39,59,73)(24,40,60,74), (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)(25,38,77,90,37,26,89,78)(27,48,79,76,39,36,91,88)(28,75,80,35,40,87,92,47)(29,34,81,86,41,46,93,74)(30,85,82,45,42,73,94,33)(31,44,83,96,43,32,95,84) );
G=PermutationGroup([[(1,41,61,75),(2,42,62,76),(3,43,63,77),(4,44,64,78),(5,45,65,79),(6,46,66,80),(7,47,67,81),(8,48,68,82),(9,25,69,83),(10,26,70,84),(11,27,71,85),(12,28,72,86),(13,29,49,87),(14,30,50,88),(15,31,51,89),(16,32,52,90),(17,33,53,91),(18,34,54,92),(19,35,55,93),(20,36,56,94),(21,37,57,95),(22,38,58,96),(23,39,59,73),(24,40,60,74)], [(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),(25,38,77,90,37,26,89,78),(27,48,79,76,39,36,91,88),(28,75,80,35,40,87,92,47),(29,34,81,86,41,46,93,74),(30,85,82,45,42,73,94,33),(31,44,83,96,43,32,95,84)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 24A | ··· | 24P |
| 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 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C8○D4 | C4○D12 | C8○D12 |
| kernel | C42.243D6 | C42.S3 | Dic3⋊C8 | D6⋊C8 | C4×C24 | C42⋊2S3 | Dic3⋊C4 | D6⋊C4 | C4×C8 | 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 | 16 |
Matrix representation of C42.243D6 ►in GL4(𝔽73) generated by
| 1 | 71 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 66 | 14 |
| 0 | 0 | 59 | 7 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 67 | 3 |
| 0 | 0 | 70 | 70 |
| 63 | 0 | 0 | 0 |
| 63 | 10 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 6 | 70 |
G:=sub<GL(4,GF(73))| [1,0,0,0,71,72,0,0,0,0,66,59,0,0,14,7],[27,0,0,0,0,27,0,0,0,0,46,0,0,0,0,46],[63,0,0,0,0,63,0,0,0,0,67,70,0,0,3,70],[63,63,0,0,0,10,0,0,0,0,3,6,0,0,3,70] >;
C42.243D6 in GAP, Magma, Sage, TeX
C_4^2._{243}D_6 % in TeX
G:=Group("C4^2.243D6"); // GroupNames label
G:=SmallGroup(192,249);
// by ID
G=gap.SmallGroup(192,249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,422,58,136,6278]);
// 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,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations