direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D7, C56⋊3C2, C8○Dic7, D14.2C4, C4.12D14, Dic7.2C4, C28.12C22, C8○(C7⋊C8), C7⋊C8⋊6C2, C7⋊1(C2×C8), C2.1(C4×D7), C14.1(C2×C4), (C4×D7).3C2, SmallGroup(112,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C8×D7 |
Generators and relations for C8×D7
G = < a,b,c | a8=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D7
►On 56 points
Generators in S56
(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 49 17 33 44 13 31)(2 50 18 34 45 14 32)(3 51 19 35 46 15 25)(4 52 20 36 47 16 26)(5 53 21 37 48 9 27)(6 54 22 38 41 10 28)(7 55 23 39 42 11 29)(8 56 24 40 43 12 30)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 37)(34 38)(35 39)(36 40)
G:=sub<Sym(56)| (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), (1,49,17,33,44,13,31)(2,50,18,34,45,14,32)(3,51,19,35,46,15,25)(4,52,20,36,47,16,26)(5,53,21,37,48,9,27)(6,54,22,38,41,10,28)(7,55,23,39,42,11,29)(8,56,24,40,43,12,30), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,37)(34,38)(35,39)(36,40)>;
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), (1,49,17,33,44,13,31)(2,50,18,34,45,14,32)(3,51,19,35,46,15,25)(4,52,20,36,47,16,26)(5,53,21,37,48,9,27)(6,54,22,38,41,10,28)(7,55,23,39,42,11,29)(8,56,24,40,43,12,30), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,37)(34,38)(35,39)(36,40) );
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)], [(1,49,17,33,44,13,31),(2,50,18,34,45,14,32),(3,51,19,35,46,15,25),(4,52,20,36,47,16,26),(5,53,21,37,48,9,27),(6,54,22,38,41,10,28),(7,55,23,39,42,11,29),(8,56,24,40,43,12,30)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(33,37),(34,38),(35,39),(36,40)]])
C8×D7 is a maximal subgroup of
C16⋊D7 D28.2C4 D28.C4 D8⋊3D7 SD16⋊3D7 Q8.D14 D21⋊C8
C8×D7 is a maximal quotient of C16⋊D7 Dic7⋊C8 D14⋊C8 D21⋊C8
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D7 | D14 | C4×D7 | C8×D7 |
| kernel | C8×D7 | C7⋊C8 | C56 | C4×D7 | Dic7 | D14 | D7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 3 | 3 | 6 | 12 |
Matrix representation of C8×D7 ►in GL2(𝔽41) generated by
| 14 | 0 |
| 0 | 14 |
| 0 | 9 |
| 9 | 37 |
| 37 | 29 |
| 32 | 4 |
G:=sub<GL(2,GF(41))| [14,0,0,14],[0,9,9,37],[37,32,29,4] >;
C8×D7 in GAP, Magma, Sage, TeX
C_8\times D_7
% in TeX
G:=Group("C8xD7"); // GroupNames label
G:=SmallGroup(112,3);
// by ID
G=gap.SmallGroup(112,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,26,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export