metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊14D4, D6⋊4SD16, C3⋊7(C8⋊8D4), C8⋊11(C3⋊D4), D6⋊3Q8⋊4C2, (C6×SD16)⋊8C2, C8⋊Dic3⋊27C2, (C2×D4).73D6, (C2×C8).263D6, (C2×Q8).78D6, (C2×SD16)⋊12S3, D6⋊3D4.8C2, C6.63(C4○D8), C12.176(C2×D4), C2.30(S3×SD16), C6.47(C2×SD16), D4⋊Dic3⋊34C2, Q8⋊2Dic3⋊29C2, (C22×S3).59D4, (C6×D4).97C22, C22.268(S3×D4), (C6×Q8).77C22, C12.101(C4○D4), C4.32(D4⋊2S3), C2.19(D6⋊3D4), C6.116(C4⋊D4), (C2×C24).164C22, (C2×C12).448C23, (C2×Dic3).114D4, C2.29(Q8.7D6), C4⋊Dic3.175C22, (S3×C2×C8)⋊8C2, C4.82(C2×C3⋊D4), (C2×C6).360(C2×D4), (C2×C3⋊C8).274C22, (S3×C2×C4).240C22, (C2×C4).537(C22×S3), SmallGroup(192,730)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊14D4
G = < a,b,c | a24=b4=c2=1, bab-1=a11, cac=a17, cbc=b-1 >
Subgroups: 360 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×C24, C3×SD16, S3×C2×C4, C2×C3⋊D4, C6×D4, C6×Q8, C8⋊8D4, C8⋊Dic3, D4⋊Dic3, Q8⋊2Dic3, S3×C2×C8, D6⋊3D4, D6⋊3Q8, C6×SD16, C24⋊14D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C4○D8, S3×D4, D4⋊2S3, C2×C3⋊D4, C8⋊8D4, S3×SD16, Q8.7D6, D6⋊3D4, C24⋊14D4
►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 39 80 56)(2 26 81 67)(3 37 82 54)(4 48 83 65)(5 35 84 52)(6 46 85 63)(7 33 86 50)(8 44 87 61)(9 31 88 72)(10 42 89 59)(11 29 90 70)(12 40 91 57)(13 27 92 68)(14 38 93 55)(15 25 94 66)(16 36 95 53)(17 47 96 64)(18 34 73 51)(19 45 74 62)(20 32 75 49)(21 43 76 60)(22 30 77 71)(23 41 78 58)(24 28 79 69)
(1 56)(2 49)(3 66)(4 59)(5 52)(6 69)(7 62)(8 55)(9 72)(10 65)(11 58)(12 51)(13 68)(14 61)(15 54)(16 71)(17 64)(18 57)(19 50)(20 67)(21 60)(22 53)(23 70)(24 63)(25 82)(26 75)(27 92)(28 85)(29 78)(30 95)(31 88)(32 81)(33 74)(34 91)(35 84)(36 77)(37 94)(38 87)(39 80)(40 73)(41 90)(42 83)(43 76)(44 93)(45 86)(46 79)(47 96)(48 89)
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), (1,39,80,56)(2,26,81,67)(3,37,82,54)(4,48,83,65)(5,35,84,52)(6,46,85,63)(7,33,86,50)(8,44,87,61)(9,31,88,72)(10,42,89,59)(11,29,90,70)(12,40,91,57)(13,27,92,68)(14,38,93,55)(15,25,94,66)(16,36,95,53)(17,47,96,64)(18,34,73,51)(19,45,74,62)(20,32,75,49)(21,43,76,60)(22,30,77,71)(23,41,78,58)(24,28,79,69), (1,56)(2,49)(3,66)(4,59)(5,52)(6,69)(7,62)(8,55)(9,72)(10,65)(11,58)(12,51)(13,68)(14,61)(15,54)(16,71)(17,64)(18,57)(19,50)(20,67)(21,60)(22,53)(23,70)(24,63)(25,82)(26,75)(27,92)(28,85)(29,78)(30,95)(31,88)(32,81)(33,74)(34,91)(35,84)(36,77)(37,94)(38,87)(39,80)(40,73)(41,90)(42,83)(43,76)(44,93)(45,86)(46,79)(47,96)(48,89)>;
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), (1,39,80,56)(2,26,81,67)(3,37,82,54)(4,48,83,65)(5,35,84,52)(6,46,85,63)(7,33,86,50)(8,44,87,61)(9,31,88,72)(10,42,89,59)(11,29,90,70)(12,40,91,57)(13,27,92,68)(14,38,93,55)(15,25,94,66)(16,36,95,53)(17,47,96,64)(18,34,73,51)(19,45,74,62)(20,32,75,49)(21,43,76,60)(22,30,77,71)(23,41,78,58)(24,28,79,69), (1,56)(2,49)(3,66)(4,59)(5,52)(6,69)(7,62)(8,55)(9,72)(10,65)(11,58)(12,51)(13,68)(14,61)(15,54)(16,71)(17,64)(18,57)(19,50)(20,67)(21,60)(22,53)(23,70)(24,63)(25,82)(26,75)(27,92)(28,85)(29,78)(30,95)(31,88)(32,81)(33,74)(34,91)(35,84)(36,77)(37,94)(38,87)(39,80)(40,73)(41,90)(42,83)(43,76)(44,93)(45,86)(46,79)(47,96)(48,89) );
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)], [(1,39,80,56),(2,26,81,67),(3,37,82,54),(4,48,83,65),(5,35,84,52),(6,46,85,63),(7,33,86,50),(8,44,87,61),(9,31,88,72),(10,42,89,59),(11,29,90,70),(12,40,91,57),(13,27,92,68),(14,38,93,55),(15,25,94,66),(16,36,95,53),(17,47,96,64),(18,34,73,51),(19,45,74,62),(20,32,75,49),(21,43,76,60),(22,30,77,71),(23,41,78,58),(24,28,79,69)], [(1,56),(2,49),(3,66),(4,59),(5,52),(6,69),(7,62),(8,55),(9,72),(10,65),(11,58),(12,51),(13,68),(14,61),(15,54),(16,71),(17,64),(18,57),(19,50),(20,67),(21,60),(22,53),(23,70),(24,63),(25,82),(26,75),(27,92),(28,85),(29,78),(30,95),(31,88),(32,81),(33,74),(34,91),(35,84),(36,77),(37,94),(38,87),(39,80),(40,73),(41,90),(42,83),(43,76),(44,93),(45,86),(46,79),(47,96),(48,89)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 2 | 3 | 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 | 6 | 6 | 8 | 2 | 2 | 2 | 6 | 6 | 8 | 24 | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | SD16 | C3⋊D4 | C4○D8 | D4⋊2S3 | S3×D4 | S3×SD16 | Q8.7D6 |
| kernel | C24⋊14D4 | C8⋊Dic3 | D4⋊Dic3 | Q8⋊2Dic3 | S3×C2×C8 | D6⋊3D4 | D6⋊3Q8 | C6×SD16 | C2×SD16 | C24 | C2×Dic3 | C22×S3 | C2×C8 | C2×D4 | C2×Q8 | C12 | D6 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C24⋊14D4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 42 |
| 0 | 0 | 40 | 12 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 27 | 30 |
| 0 | 0 | 0 | 46 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 27 | 30 |
| 0 | 0 | 39 | 46 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,8,0,0,0,0,0,40,0,0,42,12],[0,72,0,0,1,0,0,0,0,0,27,0,0,0,30,46],[0,1,0,0,1,0,0,0,0,0,27,39,0,0,30,46] >;
C24⋊14D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{14}D_4 % in TeX
G:=Group("C24:14D4"); // GroupNames label
G:=SmallGroup(192,730);
// by ID
G=gap.SmallGroup(192,730);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations