metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.1D8, C4.9D24, C42.5D6, C12.47SD16, C4⋊C8⋊2S3, C12⋊C8⋊8C2, (C2×D12).2C4, C3⋊1(C4.D8), C4.6(C24⋊C2), C4.20(D4⋊S3), (C2×C12).465D4, C4⋊D12.4C2, (C2×C4).123D12, (C4×C12).43C22, C2.4(C2.D24), C2.5(C6.D8), C6.4(C4.D4), C6.12(D4⋊C4), C22.62(D6⋊C4), C4.12(Q8⋊2S3), C2.5(C12.46D4), (C3×C4⋊C8)⋊2C2, (C2×C4).16(C4×S3), (C2×C12).28(C2×C4), (C2×C4).229(C3⋊D4), (C2×C6).47(C22⋊C4), SmallGroup(192,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D24
G = < a,b,c | a4=b24=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >
Subgroups: 376 in 84 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C12, C12, D6, C2×C6, C42, C2×C8, C2×D4, C3⋊C8, C24, D12, C2×C12, C22×S3, C4⋊C8, C4⋊C8, C4⋊1D4, C2×C3⋊C8, C4×C12, C2×C24, C2×D12, C2×D12, C4.D8, C12⋊C8, C3×C4⋊C8, C4⋊D12, C4.D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, C4.D4, D4⋊C4, C24⋊C2, D24, D6⋊C4, D4⋊S3, Q8⋊2S3, C4.D8, C6.D8, C2.D24, C12.46D4, C4.D24
►On 96 points
(1 26 49 93)(2 94 50 27)(3 28 51 95)(4 96 52 29)(5 30 53 73)(6 74 54 31)(7 32 55 75)(8 76 56 33)(9 34 57 77)(10 78 58 35)(11 36 59 79)(12 80 60 37)(13 38 61 81)(14 82 62 39)(15 40 63 83)(16 84 64 41)(17 42 65 85)(18 86 66 43)(19 44 67 87)(20 88 68 45)(21 46 69 89)(22 90 70 47)(23 48 71 91)(24 92 72 25)
(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 92 26 72 49 25 93 24)(2 23 94 48 50 71 27 91)(3 90 28 70 51 47 95 22)(4 21 96 46 52 69 29 89)(5 88 30 68 53 45 73 20)(6 19 74 44 54 67 31 87)(7 86 32 66 55 43 75 18)(8 17 76 42 56 65 33 85)(9 84 34 64 57 41 77 16)(10 15 78 40 58 63 35 83)(11 82 36 62 59 39 79 14)(12 13 80 38 60 61 37 81)
G:=sub<Sym(96)| (1,26,49,93)(2,94,50,27)(3,28,51,95)(4,96,52,29)(5,30,53,73)(6,74,54,31)(7,32,55,75)(8,76,56,33)(9,34,57,77)(10,78,58,35)(11,36,59,79)(12,80,60,37)(13,38,61,81)(14,82,62,39)(15,40,63,83)(16,84,64,41)(17,42,65,85)(18,86,66,43)(19,44,67,87)(20,88,68,45)(21,46,69,89)(22,90,70,47)(23,48,71,91)(24,92,72,25), (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), (1,92,26,72,49,25,93,24)(2,23,94,48,50,71,27,91)(3,90,28,70,51,47,95,22)(4,21,96,46,52,69,29,89)(5,88,30,68,53,45,73,20)(6,19,74,44,54,67,31,87)(7,86,32,66,55,43,75,18)(8,17,76,42,56,65,33,85)(9,84,34,64,57,41,77,16)(10,15,78,40,58,63,35,83)(11,82,36,62,59,39,79,14)(12,13,80,38,60,61,37,81)>;
G:=Group( (1,26,49,93)(2,94,50,27)(3,28,51,95)(4,96,52,29)(5,30,53,73)(6,74,54,31)(7,32,55,75)(8,76,56,33)(9,34,57,77)(10,78,58,35)(11,36,59,79)(12,80,60,37)(13,38,61,81)(14,82,62,39)(15,40,63,83)(16,84,64,41)(17,42,65,85)(18,86,66,43)(19,44,67,87)(20,88,68,45)(21,46,69,89)(22,90,70,47)(23,48,71,91)(24,92,72,25), (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), (1,92,26,72,49,25,93,24)(2,23,94,48,50,71,27,91)(3,90,28,70,51,47,95,22)(4,21,96,46,52,69,29,89)(5,88,30,68,53,45,73,20)(6,19,74,44,54,67,31,87)(7,86,32,66,55,43,75,18)(8,17,76,42,56,65,33,85)(9,84,34,64,57,41,77,16)(10,15,78,40,58,63,35,83)(11,82,36,62,59,39,79,14)(12,13,80,38,60,61,37,81) );
G=PermutationGroup([[(1,26,49,93),(2,94,50,27),(3,28,51,95),(4,96,52,29),(5,30,53,73),(6,74,54,31),(7,32,55,75),(8,76,56,33),(9,34,57,77),(10,78,58,35),(11,36,59,79),(12,80,60,37),(13,38,61,81),(14,82,62,39),(15,40,63,83),(16,84,64,41),(17,42,65,85),(18,86,66,43),(19,44,67,87),(20,88,68,45),(21,46,69,89),(22,90,70,47),(23,48,71,91),(24,92,72,25)], [(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)], [(1,92,26,72,49,25,93,24),(2,23,94,48,50,71,27,91),(3,90,28,70,51,47,95,22),(4,21,96,46,52,69,29,89),(5,88,30,68,53,45,73,20),(6,19,74,44,54,67,31,87),(7,86,32,66,55,43,75,18),(8,17,76,42,56,65,33,85),(9,84,34,64,57,41,77,16),(10,15,78,40,58,63,35,83),(11,82,36,62,59,39,79,14),(12,13,80,38,60,61,37,81)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 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 | 2 | 3 | 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 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | C24⋊C2 | D24 | C4.D4 | D4⋊S3 | Q8⋊2S3 | C12.46D4 |
| kernel | C4.D24 | C12⋊C8 | C3×C4⋊C8 | C4⋊D12 | C2×D12 | C4⋊C8 | C2×C12 | C42 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 |
Matrix representation of C4.D24 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 48 | 72 |
| 5 | 23 | 0 | 0 |
| 50 | 55 | 0 | 0 |
| 0 | 0 | 32 | 48 |
| 0 | 0 | 38 | 41 |
| 68 | 50 | 0 | 0 |
| 55 | 5 | 0 | 0 |
| 0 | 0 | 32 | 48 |
| 0 | 0 | 38 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[5,50,0,0,23,55,0,0,0,0,32,38,0,0,48,41],[68,55,0,0,50,5,0,0,0,0,32,38,0,0,48,0] >;
C4.D24 in GAP, Magma, Sage, TeX
C_4.D_{24} % in TeX
G:=Group("C4.D24"); // GroupNames label
G:=SmallGroup(192,44);
// by ID
G=gap.SmallGroup(192,44);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,100,1123,794,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations