metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.29D4, D12.27D4, Dic6.27D4, (C6×Q16)⋊2C2, C4.69(S3×D4), (C2×C8).97D6, (C2×Q16)⋊10S3, C8○D12.1C2, (C2×Q8).92D6, C24.C4⋊4C2, C12.190(C2×D4), C3⋊4(D4.5D4), C8.29(C3⋊D4), C12.10D4⋊8C2, (C2×C24).34C22, (C6×Q8).95C22, C2.24(D6⋊3D4), C6.123(C4⋊D4), (C2×C12).466C23, Q8.11D6.2C2, C4○D12.48C22, C4.Dic3.21C22, C22.22(D4⋊2S3), C4.87(C2×C3⋊D4), (C2×C6).160(C4○D4), (C2×C4).128(C22×S3), SmallGroup(192,751)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.29D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=a-1, cac=a17, cbc=b3 >
Subgroups: 248 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, S3×C8, C8⋊S3, C4.Dic3, C4.Dic3, Q8⋊2S3, C3⋊Q16, C2×C24, C3×Q16, C4○D12, C6×Q8, D4.5D4, C24.C4, C12.10D4, C8○D12, Q8.11D6, C6×Q16, C24.29D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.5D4, D6⋊3D4, C24.29D4
Character table of C24.29D4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 8 | 8 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | -√-3 | -√-3 | √-3 | √-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | √-3 | √-3 | -√-3 | -√-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | √2 | -√2 | -√-6 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√2 | √2 | -√-6 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √2 | -√2 | √-6 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | -√2 | √2 | √-6 | complex faithful |
►On 96 points
Generators in S96
(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 85 32 56 13 73 44 68)(2 84 33 55 14 96 45 67)(3 83 34 54 15 95 46 66)(4 82 35 53 16 94 47 65)(5 81 36 52 17 93 48 64)(6 80 37 51 18 92 25 63)(7 79 38 50 19 91 26 62)(8 78 39 49 20 90 27 61)(9 77 40 72 21 89 28 60)(10 76 41 71 22 88 29 59)(11 75 42 70 23 87 30 58)(12 74 43 69 24 86 31 57)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 46)(49 86)(50 79)(51 96)(52 89)(53 82)(54 75)(55 92)(56 85)(57 78)(58 95)(59 88)(60 81)(61 74)(62 91)(63 84)(64 77)(65 94)(66 87)(67 80)(68 73)(69 90)(70 83)(71 76)(72 93)
G:=sub<Sym(96)| (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), (1,85,32,56,13,73,44,68)(2,84,33,55,14,96,45,67)(3,83,34,54,15,95,46,66)(4,82,35,53,16,94,47,65)(5,81,36,52,17,93,48,64)(6,80,37,51,18,92,25,63)(7,79,38,50,19,91,26,62)(8,78,39,49,20,90,27,61)(9,77,40,72,21,89,28,60)(10,76,41,71,22,88,29,59)(11,75,42,70,23,87,30,58)(12,74,43,69,24,86,31,57), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)(49,86)(50,79)(51,96)(52,89)(53,82)(54,75)(55,92)(56,85)(57,78)(58,95)(59,88)(60,81)(61,74)(62,91)(63,84)(64,77)(65,94)(66,87)(67,80)(68,73)(69,90)(70,83)(71,76)(72,93)>;
G:=Group( (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), (1,85,32,56,13,73,44,68)(2,84,33,55,14,96,45,67)(3,83,34,54,15,95,46,66)(4,82,35,53,16,94,47,65)(5,81,36,52,17,93,48,64)(6,80,37,51,18,92,25,63)(7,79,38,50,19,91,26,62)(8,78,39,49,20,90,27,61)(9,77,40,72,21,89,28,60)(10,76,41,71,22,88,29,59)(11,75,42,70,23,87,30,58)(12,74,43,69,24,86,31,57), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)(49,86)(50,79)(51,96)(52,89)(53,82)(54,75)(55,92)(56,85)(57,78)(58,95)(59,88)(60,81)(61,74)(62,91)(63,84)(64,77)(65,94)(66,87)(67,80)(68,73)(69,90)(70,83)(71,76)(72,93) );
G=PermutationGroup([[(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)], [(1,85,32,56,13,73,44,68),(2,84,33,55,14,96,45,67),(3,83,34,54,15,95,46,66),(4,82,35,53,16,94,47,65),(5,81,36,52,17,93,48,64),(6,80,37,51,18,92,25,63),(7,79,38,50,19,91,26,62),(8,78,39,49,20,90,27,61),(9,77,40,72,21,89,28,60),(10,76,41,71,22,88,29,59),(11,75,42,70,23,87,30,58),(12,74,43,69,24,86,31,57)], [(2,18),(3,11),(5,21),(6,14),(8,24),(9,17),(12,20),(15,23),(25,45),(26,38),(27,31),(28,48),(29,41),(30,34),(32,44),(33,37),(35,47),(36,40),(39,43),(42,46),(49,86),(50,79),(51,96),(52,89),(53,82),(54,75),(55,92),(56,85),(57,78),(58,95),(59,88),(60,81),(61,74),(62,91),(63,84),(64,77),(65,94),(66,87),(67,80),(68,73),(69,90),(70,83),(71,76),(72,93)]])
Matrix representation of C24.29D4 ►in GL4(𝔽7) generated by
| 2 | 1 | 2 | 1 |
| 6 | 1 | 3 | 1 |
| 6 | 4 | 3 | 3 |
| 2 | 2 | 0 | 4 |
| 6 | 4 | 1 | 1 |
| 1 | 0 | 6 | 6 |
| 2 | 1 | 3 | 4 |
| 2 | 5 | 3 | 5 |
| 6 | 1 | 2 | 1 |
| 4 | 1 | 4 | 5 |
| 6 | 3 | 5 | 4 |
| 5 | 1 | 2 | 2 |
G:=sub<GL(4,GF(7))| [2,6,6,2,1,1,4,2,2,3,3,0,1,1,3,4],[6,1,2,2,4,0,1,5,1,6,3,3,1,6,4,5],[6,4,6,5,1,1,3,1,2,4,5,2,1,5,4,2] >;
C24.29D4 in GAP, Magma, Sage, TeX
C_{24}._{29}D_4 % in TeX
G:=Group("C24.29D4"); // GroupNames label
G:=SmallGroup(192,751);
// by ID
G=gap.SmallGroup(192,751);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,184,1123,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^3>;
// generators/relations
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