metabelian, supersoluble, monomial
Aliases: C72⋊2D4, D14⋊1D7, C14.3D14, C2.3D72, (D7×C14)⋊1C2, C7⋊2(C7⋊D4), C7⋊Dic7⋊2C2, (C7×C14).3C22, SmallGroup(392,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊2D4
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
14C2
14C2
2C7
2C7
2C7
7C22
7C22
49C4
2D7
2D7
2C14
2C14
2C14
14C14
14C14
49D4
7Dic7
7Dic7
14Dic7
14Dic7
14Dic7
Smallest permutation representation of C72⋊2D4
►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 3 5 7 2 4 6)(8 10 12 14 9 11 13)(15 20 18 16 21 19 17)(22 27 25 23 28 26 24)(29 34 32 30 35 33 31)(36 41 39 37 42 40 38)(43 45 47 49 44 46 48)(50 52 54 56 51 53 55)
(1 51 12 44)(2 50 13 43)(3 56 14 49)(4 55 8 48)(5 54 9 47)(6 53 10 46)(7 52 11 45)(15 41 22 34)(16 40 23 33)(17 39 24 32)(18 38 25 31)(19 37 26 30)(20 36 27 29)(21 42 28 35)
(1 33)(2 34)(3 35)(4 29)(5 30)(6 31)(7 32)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
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), (1,3,5,7,2,4,6)(8,10,12,14,9,11,13)(15,20,18,16,21,19,17)(22,27,25,23,28,26,24)(29,34,32,30,35,33,31)(36,41,39,37,42,40,38)(43,45,47,49,44,46,48)(50,52,54,56,51,53,55), (1,51,12,44)(2,50,13,43)(3,56,14,49)(4,55,8,48)(5,54,9,47)(6,53,10,46)(7,52,11,45)(15,41,22,34)(16,40,23,33)(17,39,24,32)(18,38,25,31)(19,37,26,30)(20,36,27,29)(21,42,28,35), (1,33)(2,34)(3,35)(4,29)(5,30)(6,31)(7,32)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)>;
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), (1,3,5,7,2,4,6)(8,10,12,14,9,11,13)(15,20,18,16,21,19,17)(22,27,25,23,28,26,24)(29,34,32,30,35,33,31)(36,41,39,37,42,40,38)(43,45,47,49,44,46,48)(50,52,54,56,51,53,55), (1,51,12,44)(2,50,13,43)(3,56,14,49)(4,55,8,48)(5,54,9,47)(6,53,10,46)(7,52,11,45)(15,41,22,34)(16,40,23,33)(17,39,24,32)(18,38,25,31)(19,37,26,30)(20,36,27,29)(21,42,28,35), (1,33)(2,34)(3,35)(4,29)(5,30)(6,31)(7,32)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56) );
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)], [(1,3,5,7,2,4,6),(8,10,12,14,9,11,13),(15,20,18,16,21,19,17),(22,27,25,23,28,26,24),(29,34,32,30,35,33,31),(36,41,39,37,42,40,38),(43,45,47,49,44,46,48),(50,52,54,56,51,53,55)], [(1,51,12,44),(2,50,13,43),(3,56,14,49),(4,55,8,48),(5,54,9,47),(6,53,10,46),(7,52,11,45),(15,41,22,34),(16,40,23,33),(17,39,24,32),(18,38,25,31),(19,37,26,30),(20,36,27,29),(21,42,28,35)], [(1,33),(2,34),(3,35),(4,29),(5,30),(6,31),(7,32),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 7G | ··· | 7O | 14A | ··· | 14F | 14G | ··· | 14O | 14P | ··· | 14AA |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 |
| size | 1 | 1 | 14 | 14 | 98 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 14 | ··· | 14 |
47 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | D4 | D7 | D14 | C7⋊D4 | D72 | C72⋊2D4 |
| kernel | C72⋊2D4 | C7⋊Dic7 | D7×C14 | C72 | D14 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 6 | 6 | 12 | 9 | 9 |
Matrix representation of C72⋊2D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 7 | 0 | 0 |
| 0 | 0 | 22 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 15 |
| 0 | 0 | 0 | 0 | 27 | 7 |
| 8 | 23 | 0 | 0 | 0 | 0 |
| 6 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 18 |
| 0 | 0 | 0 | 0 | 9 | 19 |
| 6 | 21 | 0 | 0 | 0 | 0 |
| 8 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 18 |
| 0 | 0 | 0 | 0 | 9 | 19 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,22,0,0,0,0,7,19,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,15,7],[8,6,0,0,0,0,23,21,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,18,19],[6,8,0,0,0,0,21,23,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,9,0,0,0,0,18,19] >;
C72⋊2D4 in GAP, Magma, Sage, TeX
C_7^2\rtimes_2D_4
% in TeX
G:=Group("C7^2:2D4"); // GroupNames label
G:=SmallGroup(392,20);
// by ID
G=gap.SmallGroup(392,20);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,61,488,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^7=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export