direct product, metabelian, supersoluble, monomial, A-group
Aliases: D7×C2×C14, C142⋊4C2, C72⋊2C23, C14⋊(C2×C14), C7⋊(C22×C14), (C2×C14)⋊3C14, (C7×C14)⋊2C22, SmallGroup(392,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C2×C14 |
Generators and relations for D7×C2×C14
G = < a,b,c,d | a2=b14=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 226 in 79 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C7, C7, C23, D7, C14, C14, D14, C2×C14, C2×C14, C72, C22×D7, C22×C14, C7×D7, C7×C14, D7×C14, C142, D7×C2×C14
Quotients: C1, C2, C22, C7, C23, D7, C14, D14, C2×C14, C22×D7, C22×C14, C7×D7, D7×C14, D7×C2×C14
►On 56 points
Generators in S56
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(29 53)(30 54)(31 55)(32 56)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)
(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 5 9 13 3 7 11)(2 6 10 14 4 8 12)(15 19 23 27 17 21 25)(16 20 24 28 18 22 26)(29 39 35 31 41 37 33)(30 40 36 32 42 38 34)(43 53 49 45 55 51 47)(44 54 50 46 56 52 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 43)(27 44)(28 45)
G:=sub<Sym(56)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(29,53)(30,54)(31,55)(32,56)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52), (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), (1,5,9,13,3,7,11)(2,6,10,14,4,8,12)(15,19,23,27,17,21,25)(16,20,24,28,18,22,26)(29,39,35,31,41,37,33)(30,40,36,32,42,38,34)(43,53,49,45,55,51,47)(44,54,50,46,56,52,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,43)(27,44)(28,45)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(29,53)(30,54)(31,55)(32,56)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52), (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), (1,5,9,13,3,7,11)(2,6,10,14,4,8,12)(15,19,23,27,17,21,25)(16,20,24,28,18,22,26)(29,39,35,31,41,37,33)(30,40,36,32,42,38,34)(43,53,49,45,55,51,47)(44,54,50,46,56,52,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,43)(27,44)(28,45) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(29,53),(30,54),(31,55),(32,56),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52)], [(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)], [(1,5,9,13,3,7,11),(2,6,10,14,4,8,12),(15,19,23,27,17,21,25),(16,20,24,28,18,22,26),(29,39,35,31,41,37,33),(30,40,36,32,42,38,34),(43,53,49,45,55,51,47),(44,54,50,46,56,52,48)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,29),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,43),(27,44),(28,45)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14R | 14S | ··· | 14CC | 14CD | ··· | 14DA |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D7 | D14 | C7×D7 | D7×C14 |
| kernel | D7×C2×C14 | D7×C14 | C142 | C22×D7 | D14 | C2×C14 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 6 | 1 | 6 | 36 | 6 | 3 | 9 | 18 | 54 |
Matrix representation of D7×C2×C14 ►in GL3(𝔽29) generated by
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 4 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 25 | 24 |
| 1 | 0 | 0 |
| 0 | 5 | 6 |
| 0 | 25 | 24 |
G:=sub<GL(3,GF(29))| [1,0,0,0,28,0,0,0,28],[4,0,0,0,22,0,0,0,22],[1,0,0,0,23,25,0,0,24],[1,0,0,0,5,25,0,6,24] >;
D7×C2×C14 in GAP, Magma, Sage, TeX
D_7\times C_2\times C_{14} % in TeX
G:=Group("D7xC2xC14"); // GroupNames label
G:=SmallGroup(392,42);
// by ID
G=gap.SmallGroup(392,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations