direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C4×C7⋊C3, C28⋊4C6, C14⋊2C12, (C2×C28)⋊C3, C7⋊3(C2×C12), (C2×C14).2C6, C14.6(C2×C6), C22.(C2×C7⋊C3), C2.1(C22×C7⋊C3), (C22×C7⋊C3).2C2, (C2×C7⋊C3).6C22, SmallGroup(168,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C4×C7⋊C3 |
Generators and relations for C2×C4×C7⋊C3
G = < a,b,c,d | a2=b4=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Smallest permutation representation of C2×C4×C7⋊C3
►On 56 points
Generators in S56
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)
G:=sub<Sym(56)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49), (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49), (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49)], [(1,15,8,22),(2,16,9,23),(3,17,10,24),(4,18,11,25),(5,19,12,26),(6,20,13,27),(7,21,14,28),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34),(37,38,40),(39,42,41),(44,45,47),(46,49,48),(51,52,54),(53,56,55)]])
C2×C4×C7⋊C3 is a maximal subgroup of
C28.C12 Dic7⋊C12 C28⋊C12 D14⋊C12 D28⋊6C6
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 7A | 7B | 12A | ··· | 12H | 14A | ··· | 14F | 28A | ··· | 28H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C7⋊C3 | C2×C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 |
| kernel | C2×C4×C7⋊C3 | C4×C7⋊C3 | C22×C7⋊C3 | C2×C28 | C2×C7⋊C3 | C28 | C2×C14 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 2 | 8 |
Matrix representation of C2×C4×C7⋊C3 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 148 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 |
| 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 189 |
| 1 | 0 | 0 | 0 |
| 0 | 124 | 125 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 128 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 212 | 336 | 336 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[148,0,0,0,0,189,0,0,0,0,189,0,0,0,0,189],[1,0,0,0,0,124,1,0,0,125,0,1,0,1,0,0],[128,0,0,0,0,1,212,0,0,0,336,1,0,0,336,0] >;
C2×C4×C7⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_7\rtimes C_3
% in TeX
G:=Group("C2xC4xC7:C3"); // GroupNames label
G:=SmallGroup(168,19);
// by ID
G=gap.SmallGroup(168,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,66,314]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
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