metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D7, C56⋊4C2, D14.C4, Dic7.C4, C7⋊1M4(2), C4.13D14, C28.13C22, C7⋊C8⋊4C2, C2.3(C4×D7), C14.2(C2×C4), (C4×D7).2C2, SmallGroup(112,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D7
G = < a,b,c | a8=b7=c2=1, ab=ba, cac=a5, 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 46 19 16 34 31)(2 50 47 20 9 35 32)(3 51 48 21 10 36 25)(4 52 41 22 11 37 26)(5 53 42 23 12 38 27)(6 54 43 24 13 39 28)(7 55 44 17 14 40 29)(8 56 45 18 15 33 30)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 43)(10 48)(11 45)(12 42)(13 47)(14 44)(15 41)(16 46)(18 22)(20 24)(33 52)(34 49)(35 54)(36 51)(37 56)(38 53)(39 50)(40 55)
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), (1,49,46,19,16,34,31)(2,50,47,20,9,35,32)(3,51,48,21,10,36,25)(4,52,41,22,11,37,26)(5,53,42,23,12,38,27)(6,54,43,24,13,39,28)(7,55,44,17,14,40,29)(8,56,45,18,15,33,30), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,43)(10,48)(11,45)(12,42)(13,47)(14,44)(15,41)(16,46)(18,22)(20,24)(33,52)(34,49)(35,54)(36,51)(37,56)(38,53)(39,50)(40,55)>;
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), (1,49,46,19,16,34,31)(2,50,47,20,9,35,32)(3,51,48,21,10,36,25)(4,52,41,22,11,37,26)(5,53,42,23,12,38,27)(6,54,43,24,13,39,28)(7,55,44,17,14,40,29)(8,56,45,18,15,33,30), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,43)(10,48)(11,45)(12,42)(13,47)(14,44)(15,41)(16,46)(18,22)(20,24)(33,52)(34,49)(35,54)(36,51)(37,56)(38,53)(39,50)(40,55) );
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)], [(1,49,46,19,16,34,31),(2,50,47,20,9,35,32),(3,51,48,21,10,36,25),(4,52,41,22,11,37,26),(5,53,42,23,12,38,27),(6,54,43,24,13,39,28),(7,55,44,17,14,40,29),(8,56,45,18,15,33,30)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,43),(10,48),(11,45),(12,42),(13,47),(14,44),(15,41),(16,46),(18,22),(20,24),(33,52),(34,49),(35,54),(36,51),(37,56),(38,53),(39,50),(40,55)]])
C8⋊D7 is a maximal subgroup of
D28.2C4 D7×M4(2) D28.C4 D8⋊D7 D56⋊C2 SD16⋊D7 Q16⋊D7 C8⋊F7 C28.32D6 D42.C4 C56⋊S3
C8⋊D7 is a maximal quotient of
Dic7⋊C8 C56⋊C4 D14⋊C8 C28.32D6 D42.C4 C56⋊S3
34 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 14 | 1 | 1 | 14 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D7 | M4(2) | D14 | C4×D7 | C8⋊D7 |
| kernel | C8⋊D7 | C7⋊C8 | C56 | C4×D7 | Dic7 | D14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 6 | 12 |
Matrix representation of C8⋊D7 ►in GL2(𝔽13) generated by
| 4 | 12 |
| 8 | 9 |
| 6 | 6 |
| 4 | 2 |
| 2 | 4 |
| 9 | 11 |
G:=sub<GL(2,GF(13))| [4,8,12,9],[6,4,6,2],[2,9,4,11] >;
C8⋊D7 in GAP, Magma, Sage, TeX
C_8\rtimes D_7
% in TeX
G:=Group("C8:D7"); // GroupNames label
G:=SmallGroup(112,4);
// by ID
G=gap.SmallGroup(112,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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