metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊1C8, C12.35D8, C4.17D24, C12.29SD16, C42.249D6, C12.13M4(2), (C4×C8)⋊1S3, C6.4C4≀C2, (C4×C24)⋊1C2, C3⋊1(D4⋊C8), C4.7(S3×C8), C12⋊C8⋊1C2, C12.17(C2×C8), C2.4(D6⋊C8), (C2×D12).5C4, (C4×D12).1C2, C4⋊Dic3.5C4, C4.5(C8⋊S3), (C2×C12).436D4, (C2×C4).161D12, C6.2(C22⋊C8), C4.16(C24⋊C2), C2.1(C2.D24), C6.10(D4⋊C4), C2.2(C42⋊4S3), (C4×C12).320C22, C22.31(D6⋊C4), (C2×C4).96(C4×S3), (C2×C12).216(C2×C4), (C2×C4).205(C3⋊D4), (C2×C6).36(C22⋊C4), SmallGroup(192,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.17D24
G = < a,b,c | a4=b24=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 264 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, C4⋊C8, C4×D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, D4⋊C8, C12⋊C8, C4×C24, C4×D12, C4.17D24
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), D8, SD16, C4×S3, D12, C3⋊D4, C22⋊C8, D4⋊C4, C4≀C2, S3×C8, C8⋊S3, C24⋊C2, D24, D6⋊C4, D4⋊C8, C42⋊4S3, D6⋊C8, C2.D24, C4.17D24
►On 96 points
(1 54 39 96)(2 55 40 73)(3 56 41 74)(4 57 42 75)(5 58 43 76)(6 59 44 77)(7 60 45 78)(8 61 46 79)(9 62 47 80)(10 63 48 81)(11 64 25 82)(12 65 26 83)(13 66 27 84)(14 67 28 85)(15 68 29 86)(16 69 30 87)(17 70 31 88)(18 71 32 89)(19 72 33 90)(20 49 34 91)(21 50 35 92)(22 51 36 93)(23 52 37 94)(24 53 38 95)
(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 53 54 38 39 95 96 24)(2 37 55 94 40 23 73 52)(3 93 56 22 41 51 74 36)(4 21 57 50 42 35 75 92)(5 49 58 34 43 91 76 20)(6 33 59 90 44 19 77 72)(7 89 60 18 45 71 78 32)(8 17 61 70 46 31 79 88)(9 69 62 30 47 87 80 16)(10 29 63 86 48 15 81 68)(11 85 64 14 25 67 82 28)(12 13 65 66 26 27 83 84)
G:=sub<Sym(96)| (1,54,39,96)(2,55,40,73)(3,56,41,74)(4,57,42,75)(5,58,43,76)(6,59,44,77)(7,60,45,78)(8,61,46,79)(9,62,47,80)(10,63,48,81)(11,64,25,82)(12,65,26,83)(13,66,27,84)(14,67,28,85)(15,68,29,86)(16,69,30,87)(17,70,31,88)(18,71,32,89)(19,72,33,90)(20,49,34,91)(21,50,35,92)(22,51,36,93)(23,52,37,94)(24,53,38,95), (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), (1,53,54,38,39,95,96,24)(2,37,55,94,40,23,73,52)(3,93,56,22,41,51,74,36)(4,21,57,50,42,35,75,92)(5,49,58,34,43,91,76,20)(6,33,59,90,44,19,77,72)(7,89,60,18,45,71,78,32)(8,17,61,70,46,31,79,88)(9,69,62,30,47,87,80,16)(10,29,63,86,48,15,81,68)(11,85,64,14,25,67,82,28)(12,13,65,66,26,27,83,84)>;
G:=Group( (1,54,39,96)(2,55,40,73)(3,56,41,74)(4,57,42,75)(5,58,43,76)(6,59,44,77)(7,60,45,78)(8,61,46,79)(9,62,47,80)(10,63,48,81)(11,64,25,82)(12,65,26,83)(13,66,27,84)(14,67,28,85)(15,68,29,86)(16,69,30,87)(17,70,31,88)(18,71,32,89)(19,72,33,90)(20,49,34,91)(21,50,35,92)(22,51,36,93)(23,52,37,94)(24,53,38,95), (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), (1,53,54,38,39,95,96,24)(2,37,55,94,40,23,73,52)(3,93,56,22,41,51,74,36)(4,21,57,50,42,35,75,92)(5,49,58,34,43,91,76,20)(6,33,59,90,44,19,77,72)(7,89,60,18,45,71,78,32)(8,17,61,70,46,31,79,88)(9,69,62,30,47,87,80,16)(10,29,63,86,48,15,81,68)(11,85,64,14,25,67,82,28)(12,13,65,66,26,27,83,84) );
G=PermutationGroup([[(1,54,39,96),(2,55,40,73),(3,56,41,74),(4,57,42,75),(5,58,43,76),(6,59,44,77),(7,60,45,78),(8,61,46,79),(9,62,47,80),(10,63,48,81),(11,64,25,82),(12,65,26,83),(13,66,27,84),(14,67,28,85),(15,68,29,86),(16,69,30,87),(17,70,31,88),(18,71,32,89),(19,72,33,90),(20,49,34,91),(21,50,35,92),(22,51,36,93),(23,52,37,94),(24,53,38,95)], [(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)], [(1,53,54,38,39,95,96,24),(2,37,55,94,40,23,73,52),(3,93,56,22,41,51,74,36),(4,21,57,50,42,35,75,92),(5,49,58,34,43,91,76,20),(6,33,59,90,44,19,77,72),(7,89,60,18,45,71,78,32),(8,17,61,70,46,31,79,88),(9,69,62,30,47,87,80,16),(10,29,63,86,48,15,81,68),(11,85,64,14,25,67,82,28),(12,13,65,66,26,27,83,84)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | S3×C8 | C8⋊S3 | C24⋊C2 | D24 | C42⋊4S3 |
| kernel | C4.17D24 | C12⋊C8 | C4×C24 | C4×D12 | C4⋊Dic3 | C2×D12 | D12 | C4×C8 | C2×C12 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C4.17D24 ►in GL3(𝔽73) generated by
| 27 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 63 | 0 | 0 |
| 0 | 42 | 68 |
| 0 | 5 | 37 |
| 10 | 0 | 0 |
| 0 | 5 | 37 |
| 0 | 42 | 68 |
G:=sub<GL(3,GF(73))| [27,0,0,0,46,0,0,0,46],[63,0,0,0,42,5,0,68,37],[10,0,0,0,5,42,0,37,68] >;
C4.17D24 in GAP, Magma, Sage, TeX
C_4._{17}D_{24} % in TeX
G:=Group("C4.17D24"); // GroupNames label
G:=SmallGroup(192,18);
// by ID
G=gap.SmallGroup(192,18);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,100,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations