metabelian, soluble, monomial, A-group
Aliases: C72:2C8, (C7xC14).C4, C2.(C72:C4), C7:Dic7.1C2, SmallGroup(392,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C72:2C8 |
Generators and relations for C72:2C8
G = < a,b,c | a7=b7=c8=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >
Quotients: C1, C2, C4, C8, C72:C4, C72:2C8
Smallest permutation representation of C72:2C8
►On 56 points
Generators in S56
(1 48 31 17 13 49 37)(2 50 18 41 38 14 32)(3 39 51 15 19 25 42)(4 26 16 40 43 20 52)(5 44 27 21 9 53 33)(6 54 22 45 34 10 28)(7 35 55 11 23 29 46)(8 30 12 36 47 24 56)
(1 49 17 48 37 13 31)(2 32 14 38 41 18 50)(3 25 15 39 42 19 51)(4 52 20 43 40 16 26)(5 53 21 44 33 9 27)(6 28 10 34 45 22 54)(7 29 11 35 46 23 55)(8 56 24 47 36 12 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)| (1,48,31,17,13,49,37)(2,50,18,41,38,14,32)(3,39,51,15,19,25,42)(4,26,16,40,43,20,52)(5,44,27,21,9,53,33)(6,54,22,45,34,10,28)(7,35,55,11,23,29,46)(8,30,12,36,47,24,56), (1,49,17,48,37,13,31)(2,32,14,38,41,18,50)(3,25,15,39,42,19,51)(4,52,20,43,40,16,26)(5,53,21,44,33,9,27)(6,28,10,34,45,22,54)(7,29,11,35,46,23,55)(8,56,24,47,36,12,30), (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( (1,48,31,17,13,49,37)(2,50,18,41,38,14,32)(3,39,51,15,19,25,42)(4,26,16,40,43,20,52)(5,44,27,21,9,53,33)(6,54,22,45,34,10,28)(7,35,55,11,23,29,46)(8,30,12,36,47,24,56), (1,49,17,48,37,13,31)(2,32,14,38,41,18,50)(3,25,15,39,42,19,51)(4,52,20,43,40,16,26)(5,53,21,44,33,9,27)(6,28,10,34,45,22,54)(7,29,11,35,46,23,55)(8,56,24,47,36,12,30), (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([[(1,48,31,17,13,49,37),(2,50,18,41,38,14,32),(3,39,51,15,19,25,42),(4,26,16,40,43,20,52),(5,44,27,21,9,53,33),(6,54,22,45,34,10,28),(7,35,55,11,23,29,46),(8,30,12,36,47,24,56)], [(1,49,17,48,37,13,31),(2,32,14,38,41,18,50),(3,25,15,39,42,19,51),(4,52,20,43,40,16,26),(5,53,21,44,33,9,27),(6,28,10,34,45,22,54),(7,29,11,35,46,23,55),(8,56,24,47,36,12,30)], [(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)]])
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7L | 8A | 8B | 8C | 8D | 14A | ··· | 14L |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 |
| size | 1 | 1 | 49 | 49 | 4 | ··· | 4 | 49 | 49 | 49 | 49 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||
| image | C1 | C2 | C4 | C8 | C72:C4 | C72:2C8 |
| kernel | C72:2C8 | C7:Dic7 | C7xC14 | C72 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 12 | 12 |
Matrix representation of C72:2C8 ►in GL4(F113) generated by
| 34 | 25 | 0 | 0 |
| 88 | 88 | 0 | 0 |
| 0 | 0 | 79 | 112 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 112 | 79 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 79 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 55 | 0 | 0 |
| 0 | 15 | 0 | 0 |
G:=sub<GL(4,GF(113))| [34,88,0,0,25,88,0,0,0,0,79,1,0,0,112,0],[0,112,0,0,1,79,0,0,0,0,0,112,0,0,1,79],[0,0,98,0,0,0,55,15,1,0,0,0,0,1,0,0] >;
C72:2C8 in GAP, Magma, Sage, TeX
C_7^2\rtimes_2C_8
% in TeX
G:=Group("C7^2:2C8"); // GroupNames label
G:=SmallGroup(392,17);
// by ID
G=gap.SmallGroup(392,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,7,10,26,1763,488,5004,4209]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,c*a*c^-1=a^3*b^-1,c*b*c^-1=a^3*b^4>;
// generators/relations
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