metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1D14, C28.13D4, C4.15D28, Dic28⋊2C2, M4(2)⋊2D7, C56.1C22, C22.6D28, C28.33C23, D28.8C22, Dic14.8C22, C56⋊C2⋊2C2, (C2×C14).6D4, C4○D28.4C2, (C2×C4).16D14, C14.14(C2×D4), C2.16(C2×D28), C7⋊1(C8.C22), (C2×Dic14)⋊8C2, (C7×M4(2))⋊2C2, C4.31(C22×D7), (C2×C28).28C22, SmallGroup(224,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D14
G = < a,b,c | a8=1, b14=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b13 >
Subgroups: 270 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C8.C22, C56, Dic14, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C56⋊C2, Dic28, C7×M4(2), C2×Dic14, C4○D28, C8.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, D28, C22×D7, C2×D28, C8.D14
►On 112 points
(1 83 86 33 15 69 100 47)(2 70 87 48 16 84 101 34)(3 57 88 35 17 71 102 49)(4 72 89 50 18 58 103 36)(5 59 90 37 19 73 104 51)(6 74 91 52 20 60 105 38)(7 61 92 39 21 75 106 53)(8 76 93 54 22 62 107 40)(9 63 94 41 23 77 108 55)(10 78 95 56 24 64 109 42)(11 65 96 43 25 79 110 29)(12 80 97 30 26 66 111 44)(13 67 98 45 27 81 112 31)(14 82 99 32 28 68 85 46)
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 72 43 58)(30 57 44 71)(31 70 45 84)(32 83 46 69)(33 68 47 82)(34 81 48 67)(35 66 49 80)(36 79 50 65)(37 64 51 78)(38 77 52 63)(39 62 53 76)(40 75 54 61)(41 60 55 74)(42 73 56 59)(85 100 99 86)(87 98 101 112)(88 111 102 97)(89 96 103 110)(90 109 104 95)(91 94 105 108)(92 107 106 93)
G:=sub<Sym(112)| (1,83,86,33,15,69,100,47)(2,70,87,48,16,84,101,34)(3,57,88,35,17,71,102,49)(4,72,89,50,18,58,103,36)(5,59,90,37,19,73,104,51)(6,74,91,52,20,60,105,38)(7,61,92,39,21,75,106,53)(8,76,93,54,22,62,107,40)(9,63,94,41,23,77,108,55)(10,78,95,56,24,64,109,42)(11,65,96,43,25,79,110,29)(12,80,97,30,26,66,111,44)(13,67,98,45,27,81,112,31)(14,82,99,32,28,68,85,46), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,72,43,58)(30,57,44,71)(31,70,45,84)(32,83,46,69)(33,68,47,82)(34,81,48,67)(35,66,49,80)(36,79,50,65)(37,64,51,78)(38,77,52,63)(39,62,53,76)(40,75,54,61)(41,60,55,74)(42,73,56,59)(85,100,99,86)(87,98,101,112)(88,111,102,97)(89,96,103,110)(90,109,104,95)(91,94,105,108)(92,107,106,93)>;
G:=Group( (1,83,86,33,15,69,100,47)(2,70,87,48,16,84,101,34)(3,57,88,35,17,71,102,49)(4,72,89,50,18,58,103,36)(5,59,90,37,19,73,104,51)(6,74,91,52,20,60,105,38)(7,61,92,39,21,75,106,53)(8,76,93,54,22,62,107,40)(9,63,94,41,23,77,108,55)(10,78,95,56,24,64,109,42)(11,65,96,43,25,79,110,29)(12,80,97,30,26,66,111,44)(13,67,98,45,27,81,112,31)(14,82,99,32,28,68,85,46), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,72,43,58)(30,57,44,71)(31,70,45,84)(32,83,46,69)(33,68,47,82)(34,81,48,67)(35,66,49,80)(36,79,50,65)(37,64,51,78)(38,77,52,63)(39,62,53,76)(40,75,54,61)(41,60,55,74)(42,73,56,59)(85,100,99,86)(87,98,101,112)(88,111,102,97)(89,96,103,110)(90,109,104,95)(91,94,105,108)(92,107,106,93) );
G=PermutationGroup([[(1,83,86,33,15,69,100,47),(2,70,87,48,16,84,101,34),(3,57,88,35,17,71,102,49),(4,72,89,50,18,58,103,36),(5,59,90,37,19,73,104,51),(6,74,91,52,20,60,105,38),(7,61,92,39,21,75,106,53),(8,76,93,54,22,62,107,40),(9,63,94,41,23,77,108,55),(10,78,95,56,24,64,109,42),(11,65,96,43,25,79,110,29),(12,80,97,30,26,66,111,44),(13,67,98,45,27,81,112,31),(14,82,99,32,28,68,85,46)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,72,43,58),(30,57,44,71),(31,70,45,84),(32,83,46,69),(33,68,47,82),(34,81,48,67),(35,66,49,80),(36,79,50,65),(37,64,51,78),(38,77,52,63),(39,62,53,76),(40,75,54,61),(41,60,55,74),(42,73,56,59),(85,100,99,86),(87,98,101,112),(88,111,102,97),(89,96,103,110),(90,109,104,95),(91,94,105,108),(92,107,106,93)]])
C8.D14 is a maximal subgroup of
D28.1D4 D28.2D4 D28.4D4 D28.7D4 M4(2)⋊D14 D4.9D28 C8.20D28 C8.24D28 C56.9C23 D4.11D28 D4.13D28 SD16⋊D14 D8⋊6D14 D7×C8.C22 D28.44D4
C8.D14 is a maximal quotient of
C8⋊Dic14 C42.14D14 C42.16D14 C42.20D14 C8.D28 Dic28⋊C4 C23.34D28 C23.10D28 D28.32D4 C22.D56 Dic14⋊14D4 C22⋊Dic28 Dic14.3Q8 C28⋊SD16 C42.36D14 D28⋊4Q8 C4⋊Dic28 Dic14⋊4Q8 C23.46D28 C23.47D28 C23.49D28 C56⋊2D4 C56.4D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 2 | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D28 | D28 | C8.C22 | C8.D14 |
| kernel | C8.D14 | C56⋊C2 | Dic28 | C7×M4(2) | C2×Dic14 | C4○D28 | C28 | C2×C14 | M4(2) | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 3 | 6 | 6 | 1 | 6 |
Matrix representation of C8.D14 ►in GL4(𝔽113) generated by
| 0 | 0 | 62 | 74 |
| 0 | 0 | 39 | 0 |
| 22 | 35 | 0 | 0 |
| 41 | 11 | 0 | 0 |
| 21 | 21 | 70 | 111 |
| 92 | 98 | 102 | 90 |
| 13 | 83 | 31 | 92 |
| 3 | 87 | 66 | 76 |
| 21 | 21 | 70 | 111 |
| 98 | 92 | 4 | 91 |
| 83 | 13 | 24 | 84 |
| 87 | 3 | 35 | 89 |
G:=sub<GL(4,GF(113))| [0,0,22,41,0,0,35,11,62,39,0,0,74,0,0,0],[21,92,13,3,21,98,83,87,70,102,31,66,111,90,92,76],[21,98,83,87,21,92,13,3,70,4,24,35,111,91,84,89] >;
C8.D14 in GAP, Magma, Sage, TeX
C_8.D_{14} % in TeX
G:=Group("C8.D14"); // GroupNames label
G:=SmallGroup(224,104);
// by ID
G=gap.SmallGroup(224,104);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,50,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^14=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^13>;
// generators/relations