metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.1D6, C6.5C4≀C2, C8⋊C4⋊6S3, (C2×D12).1C4, (C2×C12).223D4, (C2×C4).105D12, (C2×Dic6).1C4, C42.S3⋊1C2, C42⋊7S3.4C2, (C4×C12).10C22, C6.1(C4.D4), C2.4(D12⋊C4), C2.6(C42⋊4S3), C22.56(D6⋊C4), C2.3(C12.46D4), C3⋊1(C42.C22), (C2×C4).10(C4×S3), (C3×C8⋊C4)⋊15C2, (C2×C12).22(C2×C4), (C2×C4).206(C3⋊D4), (C2×C6).37(C22⋊C4), SmallGroup(192,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.D6
G = < a,b,c,d | a4=b4=1, c6=a-1, d2=a-1b, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=bc5 >
Subgroups: 248 in 70 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C8⋊C4, C4.4D4, C2×C3⋊C8, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×D12, C42.C22, C42.S3, C3×C8⋊C4, C42⋊7S3, C42.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.D4, C4≀C2, D6⋊C4, C42.C22, C42⋊4S3, C12.46D4, D12⋊C4, C42.D6
►On 96 points
Generators in S96
(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 81 31 67)(2 94 32 56)(3 83 33 69)(4 96 34 58)(5 85 35 71)(6 74 36 60)(7 87 37 49)(8 76 38 62)(9 89 39 51)(10 78 40 64)(11 91 41 53)(12 80 42 66)(13 93 43 55)(14 82 44 68)(15 95 45 57)(16 84 46 70)(17 73 47 59)(18 86 48 72)(19 75 25 61)(20 88 26 50)(21 77 27 63)(22 90 28 52)(23 79 29 65)(24 92 30 54)
(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 87 92 43 36 61 66)(2 79 76 35 44 53 50 17)(3 46 89 52 45 4 63 78)(5 14 91 88 47 32 65 62)(6 75 80 31 48 49 54 13)(7 42 93 72 25 24 67 74)(8 59 82 23 26 85 56 41)(9 10 95 84 27 28 69 58)(11 38 73 68 29 20 71 94)(12 55 86 19 30 81 60 37)(15 34 77 64 33 16 51 90)(21 22 83 96 39 40 57 70)
G:=sub<Sym(96)| (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), (1,81,31,67)(2,94,32,56)(3,83,33,69)(4,96,34,58)(5,85,35,71)(6,74,36,60)(7,87,37,49)(8,76,38,62)(9,89,39,51)(10,78,40,64)(11,91,41,53)(12,80,42,66)(13,93,43,55)(14,82,44,68)(15,95,45,57)(16,84,46,70)(17,73,47,59)(18,86,48,72)(19,75,25,61)(20,88,26,50)(21,77,27,63)(22,90,28,52)(23,79,29,65)(24,92,30,54), (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), (1,18,87,92,43,36,61,66)(2,79,76,35,44,53,50,17)(3,46,89,52,45,4,63,78)(5,14,91,88,47,32,65,62)(6,75,80,31,48,49,54,13)(7,42,93,72,25,24,67,74)(8,59,82,23,26,85,56,41)(9,10,95,84,27,28,69,58)(11,38,73,68,29,20,71,94)(12,55,86,19,30,81,60,37)(15,34,77,64,33,16,51,90)(21,22,83,96,39,40,57,70)>;
G:=Group( (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), (1,81,31,67)(2,94,32,56)(3,83,33,69)(4,96,34,58)(5,85,35,71)(6,74,36,60)(7,87,37,49)(8,76,38,62)(9,89,39,51)(10,78,40,64)(11,91,41,53)(12,80,42,66)(13,93,43,55)(14,82,44,68)(15,95,45,57)(16,84,46,70)(17,73,47,59)(18,86,48,72)(19,75,25,61)(20,88,26,50)(21,77,27,63)(22,90,28,52)(23,79,29,65)(24,92,30,54), (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), (1,18,87,92,43,36,61,66)(2,79,76,35,44,53,50,17)(3,46,89,52,45,4,63,78)(5,14,91,88,47,32,65,62)(6,75,80,31,48,49,54,13)(7,42,93,72,25,24,67,74)(8,59,82,23,26,85,56,41)(9,10,95,84,27,28,69,58)(11,38,73,68,29,20,71,94)(12,55,86,19,30,81,60,37)(15,34,77,64,33,16,51,90)(21,22,83,96,39,40,57,70) );
G=PermutationGroup([[(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)], [(1,81,31,67),(2,94,32,56),(3,83,33,69),(4,96,34,58),(5,85,35,71),(6,74,36,60),(7,87,37,49),(8,76,38,62),(9,89,39,51),(10,78,40,64),(11,91,41,53),(12,80,42,66),(13,93,43,55),(14,82,44,68),(15,95,45,57),(16,84,46,70),(17,73,47,59),(18,86,48,72),(19,75,25,61),(20,88,26,50),(21,77,27,63),(22,90,28,52),(23,79,29,65),(24,92,30,54)], [(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)], [(1,18,87,92,43,36,61,66),(2,79,76,35,44,53,50,17),(3,46,89,52,45,4,63,78),(5,14,91,88,47,32,65,62),(6,75,80,31,48,49,54,13),(7,42,93,72,25,24,67,74),(8,59,82,23,26,85,56,41),(9,10,95,84,27,28,69,58),(11,38,73,68,29,20,71,94),(12,55,86,19,30,81,60,37),(15,34,77,64,33,16,51,90),(21,22,83,96,39,40,57,70)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C42⋊4S3 | C4.D4 | C12.46D4 | D12⋊C4 |
| kernel | C42.D6 | C42.S3 | C3×C8⋊C4 | C42⋊7S3 | C2×Dic6 | C2×D12 | C8⋊C4 | C2×C12 | C42 | C2×C4 | C2×C4 | C2×C4 | C6 | C2 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.D6 ►in GL4(𝔽73) generated by
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 1 | 48 |
| 0 | 0 | 3 | 72 |
| 5 | 42 | 0 | 0 |
| 31 | 36 | 0 | 0 |
| 0 | 0 | 45 | 58 |
| 0 | 0 | 31 | 28 |
| 42 | 5 | 0 | 0 |
| 36 | 31 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 42 | 45 |
G:=sub<GL(4,GF(73))| [30,13,0,0,60,43,0,0,0,0,27,0,0,0,0,27],[66,14,0,0,59,7,0,0,0,0,1,3,0,0,48,72],[5,31,0,0,42,36,0,0,0,0,45,31,0,0,58,28],[42,36,0,0,5,31,0,0,0,0,0,42,0,0,15,45] >;
C42.D6 in GAP, Magma, Sage, TeX
C_4^2.D_6
% in TeX
G:=Group("C4^2.D6"); // GroupNames label
G:=SmallGroup(192,23);
// by ID
G=gap.SmallGroup(192,23);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,1571,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^-1,d^2=a^-1*b,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=b*c^5>;
// generators/relations