direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C72⋊C4, (C7×C14)⋊C4, C7⋊D7⋊2C4, C72⋊1(C2×C4), C7⋊D7.3C22, (C2×C7⋊D7).2C2, SmallGroup(392,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C2×C72⋊C4 |
Generators and relations for C2×C72⋊C4
G = < a,b,c,d | a2=b7=c7=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c-1, dcd-1=b3c4 >
49C2
49C2
2C7
2C7
2C7
2C7
49C4
49C4
49C22
2C14
2C14
2C14
2C14
14D7
14D7
14D7
14D7
14D7
14D7
14D7
14D7
49C2×C4
14D14
14D14
14D14
14D14
Permutation representations of C2×C72⋊C4
►On 28 points - transitive group 28T48
Generators in S28
(1 8)(2 10)(3 11)(4 12)(5 13)(6 14)(7 9)(15 28)(16 22)(17 23)(18 24)(19 25)(20 26)(21 27)
(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)
(1 4 7 5 10 6 3)(2 14 11 8 12 9 13)(15 18 21 17 20 16 19)(22 25 28 24 27 23 26)
(1 28 7 26)(2 18 14 17)(3 22 5 25)(4 27)(6 23 10 24)(8 15 9 20)(11 16 13 19)(12 21)
G:=sub<Sym(28)| (1,8)(2,10)(3,11)(4,12)(5,13)(6,14)(7,9)(15,28)(16,22)(17,23)(18,24)(19,25)(20,26)(21,27), (15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,4,7,5,10,6,3)(2,14,11,8,12,9,13)(15,18,21,17,20,16,19)(22,25,28,24,27,23,26), (1,28,7,26)(2,18,14,17)(3,22,5,25)(4,27)(6,23,10,24)(8,15,9,20)(11,16,13,19)(12,21)>;
G:=Group( (1,8)(2,10)(3,11)(4,12)(5,13)(6,14)(7,9)(15,28)(16,22)(17,23)(18,24)(19,25)(20,26)(21,27), (15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,4,7,5,10,6,3)(2,14,11,8,12,9,13)(15,18,21,17,20,16,19)(22,25,28,24,27,23,26), (1,28,7,26)(2,18,14,17)(3,22,5,25)(4,27)(6,23,10,24)(8,15,9,20)(11,16,13,19)(12,21) );
G=PermutationGroup([[(1,8),(2,10),(3,11),(4,12),(5,13),(6,14),(7,9),(15,28),(16,22),(17,23),(18,24),(19,25),(20,26),(21,27)], [(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)], [(1,4,7,5,10,6,3),(2,14,11,8,12,9,13),(15,18,21,17,20,16,19),(22,25,28,24,27,23,26)], [(1,28,7,26),(2,18,14,17),(3,22,5,25),(4,27),(6,23,10,24),(8,15,9,20),(11,16,13,19),(12,21)]])
G:=TransitiveGroup(28,48);
►On 28 points - transitive group 28T49
Generators in S28
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 28)(16 22)(17 23)(18 24)(19 25)(20 26)(21 27)
(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)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 18 8 24)(2 15 14 27)(3 19 13 23)(4 16 12 26)(5 20 11 22)(6 17 10 25)(7 21 9 28)
G:=sub<Sym(28)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,28)(16,22)(17,23)(18,24)(19,25)(20,26)(21,27), (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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,18,8,24)(2,15,14,27)(3,19,13,23)(4,16,12,26)(5,20,11,22)(6,17,10,25)(7,21,9,28)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,28)(16,22)(17,23)(18,24)(19,25)(20,26)(21,27), (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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,18,8,24)(2,15,14,27)(3,19,13,23)(4,16,12,26)(5,20,11,22)(6,17,10,25)(7,21,9,28) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,28),(16,22),(17,23),(18,24),(19,25),(20,26),(21,27)], [(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)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,18,8,24),(2,15,14,27),(3,19,13,23),(4,16,12,26),(5,20,11,22),(6,17,10,25),(7,21,9,28)]])
G:=TransitiveGroup(28,49);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7L | 14A | ··· | 14L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 49 | 49 | 49 | 49 | 49 | 49 | 4 | ··· | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | C72⋊C4 | C2×C72⋊C4 |
| kernel | C2×C72⋊C4 | C72⋊C4 | C2×C7⋊D7 | C7⋊D7 | C7×C14 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 12 | 12 |
Matrix representation of C2×C72⋊C4 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 |
| 28 | 3 | 0 | 0 |
| 0 | 0 | 8 | 26 |
| 0 | 0 | 3 | 28 |
| 0 | 1 | 0 | 0 |
| 28 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[0,28,0,0,1,3,0,0,0,0,8,3,0,0,26,28],[0,28,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[0,0,1,3,0,0,0,28,28,0,0,0,0,28,0,0] >;
C2×C72⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_7^2\rtimes C_4
% in TeX
G:=Group("C2xC7^2:C4"); // GroupNames label
G:=SmallGroup(392,40);
// by ID
G=gap.SmallGroup(392,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,7,20,1763,253,5004,2114]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^7=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3*c^-1,d*c*d^-1=b^3*c^4>;
// generators/relations
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