direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C8×C7⋊C3, C56⋊C3, C7⋊2C24, C28.4C6, C14.2C12, C2.(C4×C7⋊C3), C4.2(C2×C7⋊C3), (C4×C7⋊C3).4C2, (C2×C7⋊C3).2C4, SmallGroup(168,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C8×C7⋊C3 |
Generators and relations for C8×C7⋊C3
G = < a,b,c | a8=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C8×C7⋊C3
►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)
(9 50 47)(10 51 48)(11 52 41)(12 53 42)(13 54 43)(14 55 44)(15 56 45)(16 49 46)(17 29 40)(18 30 33)(19 31 34)(20 32 35)(21 25 36)(22 26 37)(23 27 38)(24 28 39)
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), (9,50,47)(10,51,48)(11,52,41)(12,53,42)(13,54,43)(14,55,44)(15,56,45)(16,49,46)(17,29,40)(18,30,33)(19,31,34)(20,32,35)(21,25,36)(22,26,37)(23,27,38)(24,28,39)>;
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), (9,50,47)(10,51,48)(11,52,41)(12,53,42)(13,54,43)(14,55,44)(15,56,45)(16,49,46)(17,29,40)(18,30,33)(19,31,34)(20,32,35)(21,25,36)(22,26,37)(23,27,38)(24,28,39) );
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)], [(9,50,47),(10,51,48),(11,52,41),(12,53,42),(13,54,43),(14,55,44),(15,56,45),(16,49,46),(17,29,40),(18,30,33),(19,31,34),(20,32,35),(21,25,36),(22,26,37),(23,27,38),(24,28,39)]])
C8×C7⋊C3 is a maximal subgroup of
C7⋊C48 C8⋊F7 C56⋊C6 D56⋊C3 C8.F7
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | 7B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | 14B | 24A | ··· | 24H | 28A | 28B | 28C | 28D | 56A | ··· | 56H |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 14 | 24 | ··· | 24 | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 3 | 3 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 3 | 3 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 | C8×C7⋊C3 |
| kernel | C8×C7⋊C3 | C4×C7⋊C3 | C56 | C2×C7⋊C3 | C28 | C7⋊C3 | C14 | C7 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 8 |
Matrix representation of C8×C7⋊C3 ►in GL3(𝔽337) generated by
| 85 | 0 | 0 |
| 0 | 85 | 0 |
| 0 | 0 | 85 |
| 212 | 213 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 124 | 336 | 336 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(337))| [85,0,0,0,85,0,0,0,85],[212,1,0,213,0,1,1,0,0],[1,124,0,0,336,1,0,336,0] >;
C8×C7⋊C3 in GAP, Magma, Sage, TeX
C_8\times C_7\rtimes C_3
% in TeX
G:=Group("C8xC7:C3"); // GroupNames label
G:=SmallGroup(168,2);
// by ID
G=gap.SmallGroup(168,2);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,609]);
// Polycyclic
G:=Group<a,b,c|a^8=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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