metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.216D6, C4⋊C4.77D6, C42.C2⋊3S3, (C2×C12).276D4, C6.110(C4○D8), C12.72(C4○D4), C6.SD16⋊41C2, C42⋊7S3.7C2, C6.D8.13C2, (C4×C12).116C22, (C2×C12).386C23, C4.14(Q8⋊3S3), C6.56(C4.4D4), C2.29(Q8.13D6), C2.9(C12.23D4), (C2×D12).104C22, C3⋊5(C42.78C22), (C2×Dic6).109C22, (C4×C3⋊C8)⋊13C2, (C2×C6).517(C2×D4), (C3×C42.C2)⋊3C2, (C2×C3⋊C8).255C22, (C2×C4).112(C3⋊D4), (C3×C4⋊C4).124C22, (C2×C4).484(C22×S3), C22.190(C2×C3⋊D4), SmallGroup(192,627)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.216D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 288 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C2×C3⋊C8, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C42.78C22, C4×C3⋊C8, C6.D8, C6.SD16, C42⋊7S3, C3×C42.C2, C42.216D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C4○D8, Q8⋊3S3, C2×C3⋊D4, C42.78C22, C12.23D4, Q8.13D6, C42.216D6
►On 96 points
Generators in S96
(1 58 35 74)(2 53 36 81)(3 60 25 76)(4 55 26 83)(5 50 27 78)(6 57 28 73)(7 52 29 80)(8 59 30 75)(9 54 31 82)(10 49 32 77)(11 56 33 84)(12 51 34 79)(13 72 48 87)(14 67 37 94)(15 62 38 89)(16 69 39 96)(17 64 40 91)(18 71 41 86)(19 66 42 93)(20 61 43 88)(21 68 44 95)(22 63 45 90)(23 70 46 85)(24 65 47 92)
(1 37 29 20)(2 21 30 38)(3 39 31 22)(4 23 32 40)(5 41 33 24)(6 13 34 42)(7 43 35 14)(8 15 36 44)(9 45 25 16)(10 17 26 46)(11 47 27 18)(12 19 28 48)(49 64 83 85)(50 86 84 65)(51 66 73 87)(52 88 74 67)(53 68 75 89)(54 90 76 69)(55 70 77 91)(56 92 78 71)(57 72 79 93)(58 94 80 61)(59 62 81 95)(60 96 82 63)
(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 6 14 48 29 34 43 19)(2 18 44 33 30 47 15 5)(3 4 16 46 31 32 45 17)(7 12 20 42 35 28 37 13)(8 24 38 27 36 41 21 11)(9 10 22 40 25 26 39 23)(49 63 91 76 83 96 70 54)(50 53 71 95 84 75 92 62)(51 61 93 74 73 94 72 52)(55 69 85 82 77 90 64 60)(56 59 65 89 78 81 86 68)(57 67 87 80 79 88 66 58)
G:=sub<Sym(96)| (1,58,35,74)(2,53,36,81)(3,60,25,76)(4,55,26,83)(5,50,27,78)(6,57,28,73)(7,52,29,80)(8,59,30,75)(9,54,31,82)(10,49,32,77)(11,56,33,84)(12,51,34,79)(13,72,48,87)(14,67,37,94)(15,62,38,89)(16,69,39,96)(17,64,40,91)(18,71,41,86)(19,66,42,93)(20,61,43,88)(21,68,44,95)(22,63,45,90)(23,70,46,85)(24,65,47,92), (1,37,29,20)(2,21,30,38)(3,39,31,22)(4,23,32,40)(5,41,33,24)(6,13,34,42)(7,43,35,14)(8,15,36,44)(9,45,25,16)(10,17,26,46)(11,47,27,18)(12,19,28,48)(49,64,83,85)(50,86,84,65)(51,66,73,87)(52,88,74,67)(53,68,75,89)(54,90,76,69)(55,70,77,91)(56,92,78,71)(57,72,79,93)(58,94,80,61)(59,62,81,95)(60,96,82,63), (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), (1,6,14,48,29,34,43,19)(2,18,44,33,30,47,15,5)(3,4,16,46,31,32,45,17)(7,12,20,42,35,28,37,13)(8,24,38,27,36,41,21,11)(9,10,22,40,25,26,39,23)(49,63,91,76,83,96,70,54)(50,53,71,95,84,75,92,62)(51,61,93,74,73,94,72,52)(55,69,85,82,77,90,64,60)(56,59,65,89,78,81,86,68)(57,67,87,80,79,88,66,58)>;
G:=Group( (1,58,35,74)(2,53,36,81)(3,60,25,76)(4,55,26,83)(5,50,27,78)(6,57,28,73)(7,52,29,80)(8,59,30,75)(9,54,31,82)(10,49,32,77)(11,56,33,84)(12,51,34,79)(13,72,48,87)(14,67,37,94)(15,62,38,89)(16,69,39,96)(17,64,40,91)(18,71,41,86)(19,66,42,93)(20,61,43,88)(21,68,44,95)(22,63,45,90)(23,70,46,85)(24,65,47,92), (1,37,29,20)(2,21,30,38)(3,39,31,22)(4,23,32,40)(5,41,33,24)(6,13,34,42)(7,43,35,14)(8,15,36,44)(9,45,25,16)(10,17,26,46)(11,47,27,18)(12,19,28,48)(49,64,83,85)(50,86,84,65)(51,66,73,87)(52,88,74,67)(53,68,75,89)(54,90,76,69)(55,70,77,91)(56,92,78,71)(57,72,79,93)(58,94,80,61)(59,62,81,95)(60,96,82,63), (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), (1,6,14,48,29,34,43,19)(2,18,44,33,30,47,15,5)(3,4,16,46,31,32,45,17)(7,12,20,42,35,28,37,13)(8,24,38,27,36,41,21,11)(9,10,22,40,25,26,39,23)(49,63,91,76,83,96,70,54)(50,53,71,95,84,75,92,62)(51,61,93,74,73,94,72,52)(55,69,85,82,77,90,64,60)(56,59,65,89,78,81,86,68)(57,67,87,80,79,88,66,58) );
G=PermutationGroup([[(1,58,35,74),(2,53,36,81),(3,60,25,76),(4,55,26,83),(5,50,27,78),(6,57,28,73),(7,52,29,80),(8,59,30,75),(9,54,31,82),(10,49,32,77),(11,56,33,84),(12,51,34,79),(13,72,48,87),(14,67,37,94),(15,62,38,89),(16,69,39,96),(17,64,40,91),(18,71,41,86),(19,66,42,93),(20,61,43,88),(21,68,44,95),(22,63,45,90),(23,70,46,85),(24,65,47,92)], [(1,37,29,20),(2,21,30,38),(3,39,31,22),(4,23,32,40),(5,41,33,24),(6,13,34,42),(7,43,35,14),(8,15,36,44),(9,45,25,16),(10,17,26,46),(11,47,27,18),(12,19,28,48),(49,64,83,85),(50,86,84,65),(51,66,73,87),(52,88,74,67),(53,68,75,89),(54,90,76,69),(55,70,77,91),(56,92,78,71),(57,72,79,93),(58,94,80,61),(59,62,81,95),(60,96,82,63)], [(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)], [(1,6,14,48,29,34,43,19),(2,18,44,33,30,47,15,5),(3,4,16,46,31,32,45,17),(7,12,20,42,35,28,37,13),(8,24,38,27,36,41,21,11),(9,10,22,40,25,26,39,23),(49,63,91,76,83,96,70,54),(50,53,71,95,84,75,92,62),(51,61,93,74,73,94,72,52),(55,69,85,82,77,90,64,60),(56,59,65,89,78,81,86,68),(57,67,87,80,79,88,66,58)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | ··· | 2 | 8 | 8 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | Q8⋊3S3 | Q8.13D6 |
| kernel | C42.216D6 | C4×C3⋊C8 | C6.D8 | C6.SD16 | C42⋊7S3 | C3×C42.C2 | C42.C2 | C2×C12 | C42 | C4⋊C4 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C42.216D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 8 |
| 0 | 0 | 0 | 0 | 55 | 46 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 3 | 0 | 0 |
| 0 | 0 | 48 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 48 | 72 |
| 30 | 43 | 0 | 0 | 0 | 0 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 48 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 18 |
| 0 | 0 | 0 | 0 | 69 | 61 |
| 43 | 30 | 0 | 0 | 0 | 0 |
| 60 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 25 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 55 |
| 0 | 0 | 0 | 0 | 4 | 12 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27,55,0,0,0,0,8,46],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,48,0,0,0,0,3,1,0,0,0,0,0,0,1,48,0,0,0,0,3,72],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,0,35,0,0,0,0,48,0,0,0,0,0,0,0,12,69,0,0,0,0,18,61],[43,60,0,0,0,0,30,30,0,0,0,0,0,0,32,35,0,0,0,0,25,0,0,0,0,0,0,0,0,4,0,0,0,0,55,12] >;
C42.216D6 in GAP, Magma, Sage, TeX
C_4^2._{216}D_6 % in TeX
G:=Group("C4^2.216D6"); // GroupNames label
G:=SmallGroup(192,627);
// by ID
G=gap.SmallGroup(192,627);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations