direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×SD16, Q8⋊C14, C8⋊2C14, C56⋊6C2, D4.C14, C14.15D4, C28.18C22, (C7×Q8)⋊4C2, C2.4(C7×D4), C4.2(C2×C14), (C7×D4).2C2, SmallGroup(112,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×SD16
G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of C7×SD16
►On 56 points
Generators in S56
(1 26 38 51 22 43 10)(2 27 39 52 23 44 11)(3 28 40 53 24 45 12)(4 29 33 54 17 46 13)(5 30 34 55 18 47 14)(6 31 35 56 19 48 15)(7 32 36 49 20 41 16)(8 25 37 50 21 42 9)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)
G:=sub<Sym(56)| (1,26,38,51,22,43,10)(2,27,39,52,23,44,11)(3,28,40,53,24,45,12)(4,29,33,54,17,46,13)(5,30,34,55,18,47,14)(6,31,35,56,19,48,15)(7,32,36,49,20,41,16)(8,25,37,50,21,42,9), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54)>;
G:=Group( (1,26,38,51,22,43,10)(2,27,39,52,23,44,11)(3,28,40,53,24,45,12)(4,29,33,54,17,46,13)(5,30,34,55,18,47,14)(6,31,35,56,19,48,15)(7,32,36,49,20,41,16)(8,25,37,50,21,42,9), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,45)(42,48)(44,46)(49,53)(50,56)(52,54) );
G=PermutationGroup([[(1,26,38,51,22,43,10),(2,27,39,52,23,44,11),(3,28,40,53,24,45,12),(4,29,33,54,17,46,13),(5,30,34,55,18,47,14),(6,31,35,56,19,48,15),(7,32,36,49,20,41,16),(8,25,37,50,21,42,9)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,45),(42,48),(44,46),(49,53),(50,56),(52,54)]])
C7×SD16 is a maximal subgroup of
D56⋊C2 SD16⋊D7 SD16⋊3D7
49 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 2 | 4 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | SD16 | C7×D4 | C7×SD16 |
| kernel | C7×SD16 | C56 | C7×D4 | C7×Q8 | SD16 | C8 | D4 | Q8 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 2 | 6 | 12 |
Matrix representation of C7×SD16 ►in GL2(𝔽43) generated by
| 16 | 0 |
| 0 | 16 |
| 8 | 2 |
| 11 | 8 |
| 42 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(43))| [16,0,0,16],[8,11,2,8],[42,0,0,1] >;
C7×SD16 in GAP, Magma, Sage, TeX
C_7\times {\rm SD}_{16} % in TeX
G:=Group("C7xSD16"); // GroupNames label
G:=SmallGroup(112,25);
// by ID
G=gap.SmallGroup(112,25);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,1683,848,58]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
Export