direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×C7⋊C8, C7⋊C56, C14.C28, C72⋊3C8, C28.8D7, C28.2C14, C14.4Dic7, C4.2(C7×D7), C2.(C7×Dic7), (C7×C28).3C2, (C7×C14).2C4, SmallGroup(392,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×C7⋊C8 |
Generators and relations for C7×C7⋊C8
G = < a,b,c | a7=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C7⋊C8
►On 56 points
Generators in S56
(1 41 19 49 31 16 39)(2 42 20 50 32 9 40)(3 43 21 51 25 10 33)(4 44 22 52 26 11 34)(5 45 23 53 27 12 35)(6 46 24 54 28 13 36)(7 47 17 55 29 14 37)(8 48 18 56 30 15 38)
(1 31 41 16 19 39 49)(2 50 40 20 9 42 32)(3 25 43 10 21 33 51)(4 52 34 22 11 44 26)(5 27 45 12 23 35 53)(6 54 36 24 13 46 28)(7 29 47 14 17 37 55)(8 56 38 18 15 48 30)
(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)
G:=sub<Sym(56)| (1,41,19,49,31,16,39)(2,42,20,50,32,9,40)(3,43,21,51,25,10,33)(4,44,22,52,26,11,34)(5,45,23,53,27,12,35)(6,46,24,54,28,13,36)(7,47,17,55,29,14,37)(8,48,18,56,30,15,38), (1,31,41,16,19,39,49)(2,50,40,20,9,42,32)(3,25,43,10,21,33,51)(4,52,34,22,11,44,26)(5,27,45,12,23,35,53)(6,54,36,24,13,46,28)(7,29,47,14,17,37,55)(8,56,38,18,15,48,30), (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)>;
G:=Group( (1,41,19,49,31,16,39)(2,42,20,50,32,9,40)(3,43,21,51,25,10,33)(4,44,22,52,26,11,34)(5,45,23,53,27,12,35)(6,46,24,54,28,13,36)(7,47,17,55,29,14,37)(8,48,18,56,30,15,38), (1,31,41,16,19,39,49)(2,50,40,20,9,42,32)(3,25,43,10,21,33,51)(4,52,34,22,11,44,26)(5,27,45,12,23,35,53)(6,54,36,24,13,46,28)(7,29,47,14,17,37,55)(8,56,38,18,15,48,30), (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) );
G=PermutationGroup([[(1,41,19,49,31,16,39),(2,42,20,50,32,9,40),(3,43,21,51,25,10,33),(4,44,22,52,26,11,34),(5,45,23,53,27,12,35),(6,46,24,54,28,13,36),(7,47,17,55,29,14,37),(8,48,18,56,30,15,38)], [(1,31,41,16,19,39,49),(2,50,40,20,9,42,32),(3,25,43,10,21,33,51),(4,52,34,22,11,44,26),(5,27,45,12,23,35,53),(6,54,36,24,13,46,28),(7,29,47,14,17,37,55),(8,56,38,18,15,48,30)], [(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)]])
140 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 7G | ··· | 7AA | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 | D7 | Dic7 | C7⋊C8 | C7×D7 | C7×Dic7 | C7×C7⋊C8 |
| kernel | C7×C7⋊C8 | C7×C28 | C7×C14 | C7⋊C8 | C72 | C28 | C14 | C7 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 | 3 | 3 | 6 | 18 | 18 | 36 |
Matrix representation of C7×C7⋊C8 ►in GL2(𝔽29) generated by
| 23 | 0 |
| 0 | 23 |
| 7 | 0 |
| 0 | 25 |
| 0 | 17 |
| 1 | 0 |
G:=sub<GL(2,GF(29))| [23,0,0,23],[7,0,0,25],[0,1,17,0] >;
C7×C7⋊C8 in GAP, Magma, Sage, TeX
C_7\times C_7\rtimes C_8
% in TeX
G:=Group("C7xC7:C8"); // GroupNames label
G:=SmallGroup(392,14);
// by ID
G=gap.SmallGroup(392,14);
# by ID
G:=PCGroup([5,-2,-7,-2,-2,-7,70,42,8404]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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