metacyclic, supersoluble, monomial
Aliases: C8⋊3F7, C56⋊4C6, D14.C12, Dic7.C12, C7⋊C8⋊4C6, C7⋊C12.C4, C8⋊D7⋊C3, C7⋊C24⋊4C2, (C2×F7).C4, C7⋊C3⋊1M4(2), C2.3(C4×F7), (C4×D7).2C6, (C4×F7).2C2, C7⋊1(C3×M4(2)), C4.13(C2×F7), C28.14(C2×C6), C14.2(C2×C12), (C8×C7⋊C3)⋊4C2, (C4×C7⋊C3).14C22, (C2×C7⋊C3).2(C2×C4), SmallGroup(336,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊F7
G = < a,b,c | a8=b7=c6=1, ab=ba, cac-1=a5, 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 34 16 19 46 49)(2 32 35 9 20 47 50)(3 25 36 10 21 48 51)(4 26 37 11 22 41 52)(5 27 38 12 23 42 53)(6 28 39 13 24 43 54)(7 29 40 14 17 44 55)(8 30 33 15 18 45 56)
(2 6)(4 8)(9 39 50 24 47 28)(10 36 51 21 48 25)(11 33 52 18 41 30)(12 38 53 23 42 27)(13 35 54 20 43 32)(14 40 55 17 44 29)(15 37 56 22 45 26)(16 34 49 19 46 31)
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), (1,31,34,16,19,46,49)(2,32,35,9,20,47,50)(3,25,36,10,21,48,51)(4,26,37,11,22,41,52)(5,27,38,12,23,42,53)(6,28,39,13,24,43,54)(7,29,40,14,17,44,55)(8,30,33,15,18,45,56), (2,6)(4,8)(9,39,50,24,47,28)(10,36,51,21,48,25)(11,33,52,18,41,30)(12,38,53,23,42,27)(13,35,54,20,43,32)(14,40,55,17,44,29)(15,37,56,22,45,26)(16,34,49,19,46,31)>;
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), (1,31,34,16,19,46,49)(2,32,35,9,20,47,50)(3,25,36,10,21,48,51)(4,26,37,11,22,41,52)(5,27,38,12,23,42,53)(6,28,39,13,24,43,54)(7,29,40,14,17,44,55)(8,30,33,15,18,45,56), (2,6)(4,8)(9,39,50,24,47,28)(10,36,51,21,48,25)(11,33,52,18,41,30)(12,38,53,23,42,27)(13,35,54,20,43,32)(14,40,55,17,44,29)(15,37,56,22,45,26)(16,34,49,19,46,31) );
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)], [(1,31,34,16,19,46,49),(2,32,35,9,20,47,50),(3,25,36,10,21,48,51),(4,26,37,11,22,41,52),(5,27,38,12,23,42,53),(6,28,39,13,24,43,54),(7,29,40,14,17,44,55),(8,30,33,15,18,45,56)], [(2,6),(4,8),(9,39,50,24,47,28),(10,36,51,21,48,25),(11,33,52,18,41,30),(12,38,53,23,42,27),(13,35,54,20,43,32),(14,40,55,17,44,29),(15,37,56,22,45,26),(16,34,49,19,46,31)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 7 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 14 | 24A | ··· | 24H | 28A | 28B | 56A | 56B | 56C | 56D |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 24 | ··· | 24 | 28 | 28 | 56 | 56 | 56 | 56 |
| size | 1 | 1 | 14 | 7 | 7 | 1 | 1 | 14 | 7 | 7 | 14 | 14 | 6 | 2 | 2 | 14 | 14 | 7 | 7 | 7 | 7 | 14 | 14 | 6 | 14 | ··· | 14 | 6 | 6 | 6 | 6 | 6 | 6 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) | 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 | Dic7 | D14 | C7⋊C3 | C7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 1 | 1 | 2 | 4 |
Matrix representation of C8⋊F7 ►in GL6(𝔽337)
| 97 | 0 | 143 | 194 | 143 | 0 |
| 194 | 97 | 143 | 0 | 0 | 143 |
| 194 | 194 | 240 | 0 | 143 | 0 |
| 0 | 194 | 0 | 97 | 143 | 143 |
| 194 | 0 | 0 | 194 | 240 | 143 |
| 0 | 194 | 143 | 194 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 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 | 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))| [97,194,194,0,194,0,0,97,194,194,0,194,143,143,240,0,0,143,194,0,0,97,194,194,143,0,143,143,240,0,0,143,0,143,143,240],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,336,336,336,336,336,336],[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\rtimes F_7
% in TeX
G:=Group("C8:F7"); // GroupNames label
G:=SmallGroup(336,8);
// by ID
G=gap.SmallGroup(336,8);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^5>;
// generators/relations
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