metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8:1D14, D56:2C2, C56:1C22, C4.14D28, C28.12D4, D28:4C22, M4(2):1D7, C22.5D28, C28.32C23, Dic14:4C22, C4oD28:2C2, (C2xD28):7C2, C56:C2:1C2, C7:1(C8:C22), (C2xC14).5D4, (C2xC4).15D14, C2.15(C2xD28), C14.13(C2xD4), (C7xM4(2)):1C2, C4.30(C22xD7), (C2xC28).27C22, SmallGroup(224,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8:D14
G = < a,b,c | a8=b14=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 398 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2xD4, C4oD4, Dic7, C28, D14, C2xC14, C8:C22, C56, Dic14, C4xD7, D28, D28, D28, C7:D4, C2xC28, C22xD7, C56:C2, D56, C7xM4(2), C2xD28, C4oD28, C8:D14
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C8:C22, D28, C22xD7, C2xD28, C8:D14
►On 56 points
(1 35 15 47 12 42 25 54)(2 29 16 55 13 36 26 48)(3 37 17 49 14 30 27 56)(4 31 18 43 8 38 28 50)(5 39 19 51 9 32 22 44)(6 33 20 45 10 40 23 52)(7 41 21 53 11 34 24 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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 21)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 28)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)
G:=sub<Sym(56)| (1,35,15,47,12,42,25,54)(2,29,16,55,13,36,26,48)(3,37,17,49,14,30,27,56)(4,31,18,43,8,38,28,50)(5,39,19,51,9,32,22,44)(6,33,20,45,10,40,23,52)(7,41,21,53,11,34,24,46), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,21)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)>;
G:=Group( (1,35,15,47,12,42,25,54)(2,29,16,55,13,36,26,48)(3,37,17,49,14,30,27,56)(4,31,18,43,8,38,28,50)(5,39,19,51,9,32,22,44)(6,33,20,45,10,40,23,52)(7,41,21,53,11,34,24,46), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,21)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,28)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50) );
G=PermutationGroup([[(1,35,15,47,12,42,25,54),(2,29,16,55,13,36,26,48),(3,37,17,49,14,30,27,56),(4,31,18,43,8,38,28,50),(5,39,19,51,9,32,22,44),(6,33,20,45,10,40,23,52),(7,41,21,53,11,34,24,46)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,21),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,28),(29,39),(30,38),(31,37),(32,36),(33,35),(40,42),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50)]])
C8:D14 is a maximal subgroup of
D28:1D4 D28.3D4 D28.5D4 D28.6D4 D4:4D28 D4.10D28 C8.21D28 C8.24D28 C56.9C23 D4.11D28 D4.12D28 D7xC8:C22 D8:5D14 D56:C22 C56.C23
C8:D14 is a maximal quotient of
C8:Dic14 C42.16D14 D56:C4 C8:D28 C42.19D14 C42.20D14 C23.35D28 D28.31D4 D28:13D4 D28:14D4 C23.38D28 C23.13D28 D28:3Q8 C4:D56 D28.19D4 D28.3Q8 Dic14:8D4 C28.7Q16 C23.47D28 C23.48D28 C23.49D28 C56:2D4 C56:3D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 28 | 2 | 2 | 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 | D56 | C7xM4(2) | C2xD28 | C4oD28 | C28 | C2xC14 | M4(2) | C8 | C2xC4 | 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
| 112 | 0 | 42 | 42 |
| 0 | 112 | 62 | 104 |
| 47 | 11 | 1 | 0 |
| 1 | 74 | 0 | 1 |
| 103 | 25 | 0 | 0 |
| 99 | 1 | 0 | 0 |
| 0 | 7 | 9 | 88 |
| 91 | 48 | 104 | 0 |
| 89 | 34 | 0 | 0 |
| 13 | 24 | 0 | 0 |
| 81 | 59 | 58 | 67 |
| 79 | 111 | 51 | 55 |
G:=sub<GL(4,GF(113))| [112,0,47,1,0,112,11,74,42,62,1,0,42,104,0,1],[103,99,0,91,25,1,7,48,0,0,9,104,0,0,88,0],[89,13,81,79,34,24,59,111,0,0,58,51,0,0,67,55] >;
C8:D14 in GAP, Magma, Sage, TeX
C_8\rtimes D_{14} % in TeX
G:=Group("C8:D14"); // GroupNames label
G:=SmallGroup(224,103);
// by ID
G=gap.SmallGroup(224,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,50,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^8=b^14=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations