direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C72⋊3C3, C72⋊10C6, (C7×C14)⋊3C3, C14⋊2(C7⋊C3), C7⋊4(C2×C7⋊C3), SmallGroup(294,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C2×C72⋊3C3 |
Generators and relations for C2×C72⋊3C3
G = < a,b,c,d | a2=b7=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b4, dcd-1=c2 >
Smallest permutation representation of C2×C72⋊3C3
►On 42 points
Generators in S42
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(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)
(1 3 5 7 2 4 6)(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)(22 24 26 28 23 25 27)(29 33 30 34 31 35 32)(36 37 38 39 40 41 42)
(1 19 10)(2 21 14)(3 16 11)(4 18 8)(5 20 12)(6 15 9)(7 17 13)(22 37 32)(23 39 29)(24 41 33)(25 36 30)(26 38 34)(27 40 31)(28 42 35)
G:=sub<Sym(42)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (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), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27)(29,33,30,34,31,35,32)(36,37,38,39,40,41,42), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13)(22,37,32)(23,39,29)(24,41,33)(25,36,30)(26,38,34)(27,40,31)(28,42,35)>;
G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (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), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27)(29,33,30,34,31,35,32)(36,37,38,39,40,41,42), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13)(22,37,32)(23,39,29)(24,41,33)(25,36,30)(26,38,34)(27,40,31)(28,42,35) );
G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)], [(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)], [(1,3,5,7,2,4,6),(8,12,9,13,10,14,11),(15,16,17,18,19,20,21),(22,24,26,28,23,25,27),(29,33,30,34,31,35,32),(36,37,38,39,40,41,42)], [(1,19,10),(2,21,14),(3,16,11),(4,18,8),(5,20,12),(6,15,9),(7,17,13),(22,37,32),(23,39,29),(24,41,33),(25,36,30),(26,38,34),(27,40,31),(28,42,35)]])
38 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7P | 14A | ··· | 14P |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 49 | 49 | 49 | 49 | 3 | ··· | 3 | 3 | ··· | 3 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7⋊C3 | C2×C7⋊C3 | C72⋊3C3 | C2×C72⋊3C3 |
| kernel | C2×C72⋊3C3 | C72⋊3C3 | C7×C14 | C72 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C2×C72⋊3C3 ►in GL3(𝔽43) generated by
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 41 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 41 | 0 |
| 0 | 0 | 16 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(43))| [42,0,0,0,42,0,0,0,42],[41,0,0,0,4,0,0,0,16],[4,0,0,0,41,0,0,0,16],[0,0,1,1,0,0,0,1,0] >;
C2×C72⋊3C3 in GAP, Magma, Sage, TeX
C_2\times C_7^2\rtimes_3C_3
% in TeX
G:=Group("C2xC7^2:3C3"); // GroupNames label
G:=SmallGroup(294,17);
// by ID
G=gap.SmallGroup(294,17);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,150,679]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^4,d*c*d^-1=c^2>;
// generators/relations
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