metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.36D4, (C6×Q16)⋊9C2, (C2×Q16)⋊8S3, (C2×C8).95D6, C8⋊Dic3⋊22C2, C3⋊5(C8.D4), C8.5(C3⋊D4), (C2×Q8).89D6, C12.187(C2×D4), D6⋊3Q8.9C2, Q8⋊2Dic3⋊36C2, (C2×Dic3).77D4, (C22×S3).41D4, C22.280(S3×D4), C12.108(C4○D4), (C6×Q8).92C22, C2.23(D6⋊3D4), C4.37(D4⋊2S3), C6.122(C4⋊D4), (C2×C24).150C22, (C2×C12).463C23, C2.30(Q16⋊S3), C6.80(C8.C22), C4⋊Dic3.186C22, C4.86(C2×C3⋊D4), (C2×C8⋊S3).5C2, (C2×C6).374(C2×D4), (S3×C2×C4).54C22, (C2×C3⋊C8).167C22, (C2×C4).551(C22×S3), SmallGroup(192,748)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.36D4
G = < a,b,c | a24=b4=c2=1, bab-1=a11, cac=a5, cbc=a12b-1 >
Subgroups: 296 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C8⋊S3, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C2×C24, C3×Q16, S3×C2×C4, C6×Q8, C8.D4, C8⋊Dic3, Q8⋊2Dic3, C2×C8⋊S3, D6⋊3Q8, C6×Q16, C24.36D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C8.C22, S3×D4, D4⋊2S3, C2×C3⋊D4, C8.D4, Q16⋊S3, D6⋊3D4, C24.36D4
Character table of C24.36D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 8 | 8 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 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 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | -1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 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 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | √-6 | -√-6 | -√-6 | complex lifted from Q16⋊S3 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | -√-6 | √-6 | √-6 | complex lifted from Q16⋊S3 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | √-6 | -√-6 | complex lifted from Q16⋊S3 |
| ρ30 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | -√-6 | √-6 | complex lifted from Q16⋊S3 |
►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 88 38 63)(2 75 39 50)(3 86 40 61)(4 73 41 72)(5 84 42 59)(6 95 43 70)(7 82 44 57)(8 93 45 68)(9 80 46 55)(10 91 47 66)(11 78 48 53)(12 89 25 64)(13 76 26 51)(14 87 27 62)(15 74 28 49)(16 85 29 60)(17 96 30 71)(18 83 31 58)(19 94 32 69)(20 81 33 56)(21 92 34 67)(22 79 35 54)(23 90 36 65)(24 77 37 52)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 45)(27 31)(28 36)(29 41)(30 46)(33 37)(34 42)(35 47)(39 43)(40 48)(49 78)(50 83)(51 88)(52 93)(53 74)(54 79)(55 84)(56 89)(57 94)(58 75)(59 80)(60 85)(61 90)(62 95)(63 76)(64 81)(65 86)(66 91)(67 96)(68 77)(69 82)(70 87)(71 92)(72 73)
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), (1,88,38,63)(2,75,39,50)(3,86,40,61)(4,73,41,72)(5,84,42,59)(6,95,43,70)(7,82,44,57)(8,93,45,68)(9,80,46,55)(10,91,47,66)(11,78,48,53)(12,89,25,64)(13,76,26,51)(14,87,27,62)(15,74,28,49)(16,85,29,60)(17,96,30,71)(18,83,31,58)(19,94,32,69)(20,81,33,56)(21,92,34,67)(22,79,35,54)(23,90,36,65)(24,77,37,52), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48)(49,78)(50,83)(51,88)(52,93)(53,74)(54,79)(55,84)(56,89)(57,94)(58,75)(59,80)(60,85)(61,90)(62,95)(63,76)(64,81)(65,86)(66,91)(67,96)(68,77)(69,82)(70,87)(71,92)(72,73)>;
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), (1,88,38,63)(2,75,39,50)(3,86,40,61)(4,73,41,72)(5,84,42,59)(6,95,43,70)(7,82,44,57)(8,93,45,68)(9,80,46,55)(10,91,47,66)(11,78,48,53)(12,89,25,64)(13,76,26,51)(14,87,27,62)(15,74,28,49)(16,85,29,60)(17,96,30,71)(18,83,31,58)(19,94,32,69)(20,81,33,56)(21,92,34,67)(22,79,35,54)(23,90,36,65)(24,77,37,52), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48)(49,78)(50,83)(51,88)(52,93)(53,74)(54,79)(55,84)(56,89)(57,94)(58,75)(59,80)(60,85)(61,90)(62,95)(63,76)(64,81)(65,86)(66,91)(67,96)(68,77)(69,82)(70,87)(71,92)(72,73) );
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)], [(1,88,38,63),(2,75,39,50),(3,86,40,61),(4,73,41,72),(5,84,42,59),(6,95,43,70),(7,82,44,57),(8,93,45,68),(9,80,46,55),(10,91,47,66),(11,78,48,53),(12,89,25,64),(13,76,26,51),(14,87,27,62),(15,74,28,49),(16,85,29,60),(17,96,30,71),(18,83,31,58),(19,94,32,69),(20,81,33,56),(21,92,34,67),(22,79,35,54),(23,90,36,65),(24,77,37,52)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,45),(27,31),(28,36),(29,41),(30,46),(33,37),(34,42),(35,47),(39,43),(40,48),(49,78),(50,83),(51,88),(52,93),(53,74),(54,79),(55,84),(56,89),(57,94),(58,75),(59,80),(60,85),(61,90),(62,95),(63,76),(64,81),(65,86),(66,91),(67,96),(68,77),(69,82),(70,87),(71,92),(72,73)]])
Matrix representation of C24.36D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 42 | 62 | 42 |
| 0 | 0 | 31 | 31 | 31 | 31 |
| 0 | 0 | 11 | 31 | 62 | 42 |
| 0 | 0 | 42 | 42 | 31 | 31 |
| 17 | 70 | 0 | 0 | 0 | 0 |
| 48 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 41 | 17 | 56 |
| 0 | 0 | 0 | 41 | 0 | 56 |
| 0 | 0 | 17 | 56 | 41 | 32 |
| 0 | 0 | 0 | 56 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,62,31,11,42,0,0,42,31,31,42,0,0,62,31,62,31,0,0,42,31,42,31],[17,48,0,0,0,0,70,56,0,0,0,0,0,0,32,0,17,0,0,0,41,41,56,56,0,0,17,0,41,0,0,0,56,56,32,32],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;
C24.36D4 in GAP, Magma, Sage, TeX
C_{24}._{36}D_4 % in TeX
G:=Group("C24.36D4"); // GroupNames label
G:=SmallGroup(192,748);
// by ID
G=gap.SmallGroup(192,748);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,219,184,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^5,c*b*c=a^12*b^-1>;
// generators/relations
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