direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D7×C14, C14⋊C14, C72⋊2C22, C7⋊(C2×C14), (C7×C14)⋊1C2, SmallGroup(196,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C14 |
Generators and relations for D7×C14
G = < a,b,c | a14=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of D7×C14
►On 28 points - transitive group 28T34
Generators in S28
(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 13 11 9 7 5 3)(2 14 12 10 8 6 4)(15 17 19 21 23 25 27)(16 18 20 22 24 26 28)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)
G:=sub<Sym(28)| (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), (1,13,11,9,7,5,3)(2,14,12,10,8,6,4)(15,17,19,21,23,25,27)(16,18,20,22,24,26,28), (1,24)(2,25)(3,26)(4,27)(5,28)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)>;
G:=Group( (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), (1,13,11,9,7,5,3)(2,14,12,10,8,6,4)(15,17,19,21,23,25,27)(16,18,20,22,24,26,28), (1,24)(2,25)(3,26)(4,27)(5,28)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23) );
G=PermutationGroup([[(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)], [(1,13,11,9,7,5,3),(2,14,12,10,8,6,4),(15,17,19,21,23,25,27),(16,18,20,22,24,26,28)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23)]])
G:=TransitiveGroup(28,34);
D7×C14 is a maximal subgroup of
C72⋊2D4 C7⋊D28
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 14G | ··· | 14AA | 14AB | ··· | 14AM |
| order | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 |
| size | 1 | 1 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
70 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×C14 | C7×D7 | C7×C14 | D14 | D7 | C14 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 3 | 3 | 18 | 18 |
Matrix representation of D7×C14 ►in GL2(𝔽29) generated by
| 5 | 0 |
| 0 | 5 |
| 16 | 0 |
| 0 | 20 |
| 0 | 20 |
| 16 | 0 |
G:=sub<GL(2,GF(29))| [5,0,0,5],[16,0,0,20],[0,16,20,0] >;
D7×C14 in GAP, Magma, Sage, TeX
D_7\times C_{14} % in TeX
G:=Group("D7xC14"); // GroupNames label
G:=SmallGroup(196,10);
// by ID
G=gap.SmallGroup(196,10);
# by ID
G:=PCGroup([4,-2,-2,-7,-7,2691]);
// Polycyclic
G:=Group<a,b,c|a^14=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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