direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×C7⋊C3, C7⋊C21, C72⋊1C3, SmallGroup(147,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×C7⋊C3 |
Generators and relations for C7×C7⋊C3
G = < a,b,c | a7=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Permutation representations of C7×C7⋊C3
►On 21 points - transitive group 21T13
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)
G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,2,3,4,5,6,7),(8,10,12,14,9,11,13),(15,19,16,20,17,21,18)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)]])
G:=TransitiveGroup(21,13);
C7×C7⋊C3 is a maximal subgroup of
C7⋊5F7
35 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7F | 7G | ··· | 7T | 21A | ··· | 21L |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 7 | 7 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C7 | C21 | C7⋊C3 | C7×C7⋊C3 |
| kernel | C7×C7⋊C3 | C72 | C7⋊C3 | C7 | C7 | C1 |
| # reps | 1 | 2 | 6 | 12 | 2 | 12 |
Matrix representation of C7×C7⋊C3 ►in GL3(𝔽43) generated by
| 35 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 41 | 0 | 0 |
| 0 | 4 | 0 |
| 2 | 5 | 16 |
| 0 | 1 | 0 |
| 7 | 37 | 20 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(43))| [35,0,0,0,35,0,0,0,35],[41,0,2,0,4,5,0,0,16],[0,7,0,1,37,0,0,20,6] >;
C7×C7⋊C3 in GAP, Magma, Sage, TeX
C_7\times C_7\rtimes C_3
% in TeX
G:=Group("C7xC7:C3"); // GroupNames label
G:=SmallGroup(147,3);
// by ID
G=gap.SmallGroup(147,3);
# by ID
G:=PCGroup([3,-3,-7,-7,380]);
// Polycyclic
G:=Group<a,b,c|a^7=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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