direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C8×F7, D7⋊C24, C56⋊3C6, D14.2C12, Dic7.2C12, (C8×D7)⋊C3, C7⋊C8⋊6C6, C7⋊C24⋊6C2, C7⋊1(C2×C24), C7⋊C12.2C4, C2.1(C4×F7), (C4×D7).3C6, (C4×F7).3C2, (C2×F7).2C4, C4.12(C2×F7), C14.1(C2×C12), C28.13(C2×C6), C7⋊C3⋊1(C2×C8), (C8×C7⋊C3)⋊3C2, (C4×C7⋊C3).13C22, (C2×C7⋊C3).1(C2×C4), SmallGroup(336,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C8×F7 |
Generators and relations for C8×F7
G = < a,b,c | a8=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C8×F7
►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 31 13 37 48 17 49)(2 32 14 38 41 18 50)(3 25 15 39 42 19 51)(4 26 16 40 43 20 52)(5 27 9 33 44 21 53)(6 28 10 34 45 22 54)(7 29 11 35 46 23 55)(8 30 12 36 47 24 56)
(9 53 44 21 27 33)(10 54 45 22 28 34)(11 55 46 23 29 35)(12 56 47 24 30 36)(13 49 48 17 31 37)(14 50 41 18 32 38)(15 51 42 19 25 39)(16 52 43 20 26 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), (1,31,13,37,48,17,49)(2,32,14,38,41,18,50)(3,25,15,39,42,19,51)(4,26,16,40,43,20,52)(5,27,9,33,44,21,53)(6,28,10,34,45,22,54)(7,29,11,35,46,23,55)(8,30,12,36,47,24,56), (9,53,44,21,27,33)(10,54,45,22,28,34)(11,55,46,23,29,35)(12,56,47,24,30,36)(13,49,48,17,31,37)(14,50,41,18,32,38)(15,51,42,19,25,39)(16,52,43,20,26,40)>;
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), (1,31,13,37,48,17,49)(2,32,14,38,41,18,50)(3,25,15,39,42,19,51)(4,26,16,40,43,20,52)(5,27,9,33,44,21,53)(6,28,10,34,45,22,54)(7,29,11,35,46,23,55)(8,30,12,36,47,24,56), (9,53,44,21,27,33)(10,54,45,22,28,34)(11,55,46,23,29,35)(12,56,47,24,30,36)(13,49,48,17,31,37)(14,50,41,18,32,38)(15,51,42,19,25,39)(16,52,43,20,26,40) );
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)], [(1,31,13,37,48,17,49),(2,32,14,38,41,18,50),(3,25,15,39,42,19,51),(4,26,16,40,43,20,52),(5,27,9,33,44,21,53),(6,28,10,34,45,22,54),(7,29,11,35,46,23,55),(8,30,12,36,47,24,56)], [(9,53,44,21,27,33),(10,54,45,22,28,34),(11,55,46,23,29,35),(12,56,47,24,30,36),(13,49,48,17,31,37),(14,50,41,18,32,38),(15,51,42,19,25,39),(16,52,43,20,26,40)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 7 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 14 | 24A | ··· | 24P | 28A | 28B | 56A | 56B | 56C | 56D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 14 | 24 | ··· | 24 | 28 | 28 | 56 | 56 | 56 | 56 |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 7 | 7 | 7 | ··· | 7 | 6 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | ··· | 7 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 | 6 | 6 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | F7 | C2×F7 | C4×F7 | C8×F7 |
| kernel | C8×F7 | C7⋊C24 | C8×C7⋊C3 | C4×F7 | C8×D7 | C7⋊C12 | C2×F7 | C7⋊C8 | C56 | C4×D7 | F7 | Dic7 | D14 | D7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C8×F7 ►in GL6(𝔽337)
| 85 | 0 | 0 | 0 | 0 | 0 |
| 0 | 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 85 | 0 | 0 | 0 |
| 0 | 0 | 0 | 85 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 0 |
| 0 | 0 | 0 | 0 | 0 | 85 |
| 336 | 336 | 336 | 336 | 336 | 336 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(337))| [85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C8×F7 in GAP, Magma, Sage, TeX
C_8\times F_7
% in TeX
G:=Group("C8xF7"); // GroupNames label
G:=SmallGroup(336,7);
// by ID
G=gap.SmallGroup(336,7);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,79,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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