direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×Dic7, C7⋊C28, C14.C14, C72⋊2C4, C14.4D7, C2.(C7×D7), (C7×C14).1C2, SmallGroup(196,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×Dic7 |
Generators and relations for C7×Dic7
G = < a,b,c | a7=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Permutation representations of C7×Dic7
►On 28 points - transitive group 28T33
Generators in S28
(1 5 9 13 3 7 11)(2 6 10 14 4 8 12)(15 25 21 17 27 23 19)(16 26 22 18 28 24 20)
(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 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)
G:=sub<Sym(28)| (1,5,9,13,3,7,11)(2,6,10,14,4,8,12)(15,25,21,17,27,23,19)(16,26,22,18,28,24,20), (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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23)>;
G:=Group( (1,5,9,13,3,7,11)(2,6,10,14,4,8,12)(15,25,21,17,27,23,19)(16,26,22,18,28,24,20), (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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23) );
G=PermutationGroup([[(1,5,9,13,3,7,11),(2,6,10,14,4,8,12),(15,25,21,17,27,23,19),(16,26,22,18,28,24,20)], [(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)], [(1,22,8,15),(2,21,9,28),(3,20,10,27),(4,19,11,26),(5,18,12,25),(6,17,13,24),(7,16,14,23)]])
G:=TransitiveGroup(28,33);
C7×Dic7 is a maximal subgroup of
Dic7⋊2D7 C7⋊D28 C72⋊2Q8 D7×C28
70 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 14G | ··· | 14AA | 28A | ··· | 28L |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| 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 | C4 | C7 | C14 | C28 | D7 | Dic7 | C7×D7 | C7×Dic7 |
| kernel | C7×Dic7 | C7×C14 | C72 | Dic7 | C14 | C7 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 3 | 3 | 18 | 18 |
Matrix representation of C7×Dic7 ►in GL2(𝔽29) generated by
| 24 | 0 |
| 0 | 24 |
| 22 | 0 |
| 0 | 4 |
| 0 | 28 |
| 1 | 0 |
G:=sub<GL(2,GF(29))| [24,0,0,24],[22,0,0,4],[0,1,28,0] >;
C7×Dic7 in GAP, Magma, Sage, TeX
C_7\times {\rm Dic}_7 % in TeX
G:=Group("C7xDic7"); // GroupNames label
G:=SmallGroup(196,5);
// by ID
G=gap.SmallGroup(196,5);
# by ID
G:=PCGroup([4,-2,-7,-2,-7,56,2691]);
// Polycyclic
G:=Group<a,b,c|a^7=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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