metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.43D4, C3⋊C8.20D4, C4.24(S3×D4), (C6×SD16)⋊7C2, (C2×SD16)⋊9S3, (C2×D4).71D6, (C2×C8).262D6, (C2×Q8).77D6, (C8×Dic3)⋊10C2, C6.62(C4○D8), C12.175(C2×D4), C8.20(C3⋊D4), C3⋊4(C8.12D4), C23.12D6⋊6C2, C12.23D4⋊4C2, C6.30(C4⋊1D4), (C6×D4).94C22, C22.265(S3×D4), (C6×Q8).75C22, C2.21(C12⋊3D4), (C2×C12).445C23, (C2×C24).163C22, (C2×Dic3).113D4, C2.28(Q8.7D6), (C2×D12).119C22, (C2×Dic6).126C22, (C4×Dic3).241C22, C4.8(C2×C3⋊D4), (C2×D4⋊S3).9C2, (C2×C24⋊C2)⋊29C2, (C2×C3⋊Q16)⋊18C2, (C2×C6).357(C2×D4), (C2×C3⋊C8).273C22, (C2×C4).534(C22×S3), SmallGroup(192,727)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.43D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a11, cbc-1=a12b-1 >
Subgroups: 408 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×SD16, C2×Q16, C24⋊C2, C2×C3⋊C8, C4×Dic3, D6⋊C4, D4⋊S3, C3⋊Q16, C6.D4, C2×C24, C3×SD16, C2×Dic6, C2×D12, C6×D4, C6×Q8, C8.12D4, C8×Dic3, C2×C24⋊C2, C2×D4⋊S3, C23.12D6, C2×C3⋊Q16, C12.23D4, C6×SD16, C24.43D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C4○D8, S3×D4, C2×C3⋊D4, C8.12D4, Q8.7D6, C12⋊3D4, C24.43D4
►On 96 points
(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 93 56 34)(2 86 57 27)(3 79 58 44)(4 96 59 37)(5 89 60 30)(6 82 61 47)(7 75 62 40)(8 92 63 33)(9 85 64 26)(10 78 65 43)(11 95 66 36)(12 88 67 29)(13 81 68 46)(14 74 69 39)(15 91 70 32)(16 84 71 25)(17 77 72 42)(18 94 49 35)(19 87 50 28)(20 80 51 45)(21 73 52 38)(22 90 53 31)(23 83 54 48)(24 76 55 41)
(1 16 13 4)(2 3 14 15)(5 12 17 24)(6 23 18 11)(7 10 19 22)(8 21 20 9)(25 93 37 81)(26 80 38 92)(27 91 39 79)(28 78 40 90)(29 89 41 77)(30 76 42 88)(31 87 43 75)(32 74 44 86)(33 85 45 73)(34 96 46 84)(35 83 47 95)(36 94 48 82)(49 66 61 54)(50 53 62 65)(51 64 63 52)(55 60 67 72)(56 71 68 59)(57 58 69 70)
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), (1,93,56,34)(2,86,57,27)(3,79,58,44)(4,96,59,37)(5,89,60,30)(6,82,61,47)(7,75,62,40)(8,92,63,33)(9,85,64,26)(10,78,65,43)(11,95,66,36)(12,88,67,29)(13,81,68,46)(14,74,69,39)(15,91,70,32)(16,84,71,25)(17,77,72,42)(18,94,49,35)(19,87,50,28)(20,80,51,45)(21,73,52,38)(22,90,53,31)(23,83,54,48)(24,76,55,41), (1,16,13,4)(2,3,14,15)(5,12,17,24)(6,23,18,11)(7,10,19,22)(8,21,20,9)(25,93,37,81)(26,80,38,92)(27,91,39,79)(28,78,40,90)(29,89,41,77)(30,76,42,88)(31,87,43,75)(32,74,44,86)(33,85,45,73)(34,96,46,84)(35,83,47,95)(36,94,48,82)(49,66,61,54)(50,53,62,65)(51,64,63,52)(55,60,67,72)(56,71,68,59)(57,58,69,70)>;
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), (1,93,56,34)(2,86,57,27)(3,79,58,44)(4,96,59,37)(5,89,60,30)(6,82,61,47)(7,75,62,40)(8,92,63,33)(9,85,64,26)(10,78,65,43)(11,95,66,36)(12,88,67,29)(13,81,68,46)(14,74,69,39)(15,91,70,32)(16,84,71,25)(17,77,72,42)(18,94,49,35)(19,87,50,28)(20,80,51,45)(21,73,52,38)(22,90,53,31)(23,83,54,48)(24,76,55,41), (1,16,13,4)(2,3,14,15)(5,12,17,24)(6,23,18,11)(7,10,19,22)(8,21,20,9)(25,93,37,81)(26,80,38,92)(27,91,39,79)(28,78,40,90)(29,89,41,77)(30,76,42,88)(31,87,43,75)(32,74,44,86)(33,85,45,73)(34,96,46,84)(35,83,47,95)(36,94,48,82)(49,66,61,54)(50,53,62,65)(51,64,63,52)(55,60,67,72)(56,71,68,59)(57,58,69,70) );
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)], [(1,93,56,34),(2,86,57,27),(3,79,58,44),(4,96,59,37),(5,89,60,30),(6,82,61,47),(7,75,62,40),(8,92,63,33),(9,85,64,26),(10,78,65,43),(11,95,66,36),(12,88,67,29),(13,81,68,46),(14,74,69,39),(15,91,70,32),(16,84,71,25),(17,77,72,42),(18,94,49,35),(19,87,50,28),(20,80,51,45),(21,73,52,38),(22,90,53,31),(23,83,54,48),(24,76,55,41)], [(1,16,13,4),(2,3,14,15),(5,12,17,24),(6,23,18,11),(7,10,19,22),(8,21,20,9),(25,93,37,81),(26,80,38,92),(27,91,39,79),(28,78,40,90),(29,89,41,77),(30,76,42,88),(31,87,43,75),(32,74,44,86),(33,85,45,73),(34,96,46,84),(35,83,47,95),(36,94,48,82),(49,66,61,54),(50,53,62,65),(51,64,63,52),(55,60,67,72),(56,71,68,59),(57,58,69,70)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C4○D8 | S3×D4 | S3×D4 | Q8.7D6 |
| kernel | C24.43D4 | C8×Dic3 | C2×C24⋊C2 | C2×D4⋊S3 | C23.12D6 | C2×C3⋊Q16 | C12.23D4 | C6×SD16 | C2×SD16 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×D4 | C2×Q8 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 4 |
Matrix representation of C24.43D4 ►in GL6(𝔽73)
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 71 |
| 0 | 0 | 0 | 0 | 18 | 20 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 2 |
| 0 | 0 | 0 | 0 | 55 | 53 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 71 |
| 0 | 0 | 0 | 0 | 17 | 20 |
G:=sub<GL(6,GF(73))| [67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,53,18,0,0,0,0,71,20],[0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,55,0,0,0,0,2,53],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,53,17,0,0,0,0,71,20] >;
C24.43D4 in GAP, Magma, Sage, TeX
C_{24}._{43}D_4 % in TeX
G:=Group("C24.43D4"); // GroupNames label
G:=SmallGroup(192,727);
// by ID
G=gap.SmallGroup(192,727);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^17,c*a*c^-1=a^11,c*b*c^-1=a^12*b^-1>;
// generators/relations