direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4, C4⋊C14, C28⋊3C2, C22⋊C14, C14.6C22, (C2×C14)⋊1C2, C2.1(C2×C14), SmallGroup(56,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4
G = < a,b,c | a7=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C7×D4
►On 28 points - transitive group 28T5
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 12 19 24)(2 13 20 25)(3 14 21 26)(4 8 15 27)(5 9 16 28)(6 10 17 22)(7 11 18 23)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
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), (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)>;
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), (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21) );
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)], [(1,12,19,24),(2,13,20,25),(3,14,21,26),(4,8,15,27),(5,9,16,28),(6,10,17,22),(7,11,18,23)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)]])
G:=TransitiveGroup(28,5);
C7×D4 is a maximal subgroup of
D4⋊D7 D4.D7 D4⋊2D7
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14R | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D4 | C7×D4 |
| kernel | C7×D4 | C28 | C2×C14 | D4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 6 |
Matrix representation of C7×D4 ►in GL2(𝔽29) generated by
| 24 | 0 |
| 0 | 24 |
| 28 | 1 |
| 27 | 1 |
| 1 | 28 |
| 0 | 28 |
G:=sub<GL(2,GF(29))| [24,0,0,24],[28,27,1,1],[1,0,28,28] >;
C7×D4 in GAP, Magma, Sage, TeX
C_7\times D_4
% in TeX
G:=Group("C7xD4"); // GroupNames label
G:=SmallGroup(56,9);
// by ID
G=gap.SmallGroup(56,9);
# by ID
G:=PCGroup([4,-2,-2,-7,-2,241]);
// Polycyclic
G:=Group<a,b,c|a^7=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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