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G = C14xDic7order 392 = 23·72

Direct product of C14 and Dic7

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C14xDic7, C14:C28, C14.20D14, C142.1C2, C7:2(C2xC28), (C7xC14):2C4, C72:6(C2xC4), C22.(C7xD7), (C2xC14).5D7, C2.2(D7xC14), (C2xC14).2C14, C14.4(C2xC14), (C7xC14).9C22, SmallGroup(392,26)

Series: Derived Chief Lower central Upper central

C1C7 — C14xDic7
C1C7C14C7xC14C7xDic7 — C14xDic7
C7 — C14xDic7
C1C2xC14

Generators and relations for C14xDic7
 G = < a,b,c | a14=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 98 in 47 conjugacy classes, 26 normal (14 characteristic)
Quotients: C1, C2, C4, C22, C7, C2xC4, D7, C14, Dic7, C28, D14, C2xC14, C2xDic7, C2xC28, C7xD7, C7xDic7, D7xC14, C14xDic7
2C7
2C7
2C7
7C4
7C4
2C14
2C14
2C14
2C14
2C14
2C14
2C14
2C14
2C14
7C2xC4
2C2xC14
2C2xC14
2C2xC14
7C28
7C28
7C2xC28

Smallest permutation representation of C14xDic7
On 56 points
Generators in S56
(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)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 44 9 52 3 46 11 54 5 48 13 56 7 50)(2 45 10 53 4 47 12 55 6 49 14 43 8 51)(15 32 21 38 27 30 19 36 25 42 17 34 23 40)(16 33 22 39 28 31 20 37 26 29 18 35 24 41)
(1 16 54 37)(2 17 55 38)(3 18 56 39)(4 19 43 40)(5 20 44 41)(6 21 45 42)(7 22 46 29)(8 23 47 30)(9 24 48 31)(10 25 49 32)(11 26 50 33)(12 27 51 34)(13 28 52 35)(14 15 53 36)

G:=sub<Sym(56)| (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36)>;

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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36) );

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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,44,9,52,3,46,11,54,5,48,13,56,7,50),(2,45,10,53,4,47,12,55,6,49,14,43,8,51),(15,32,21,38,27,30,19,36,25,42,17,34,23,40),(16,33,22,39,28,31,20,37,26,29,18,35,24,41)], [(1,16,54,37),(2,17,55,38),(3,18,56,39),(4,19,43,40),(5,20,44,41),(6,21,45,42),(7,22,46,29),(8,23,47,30),(9,24,48,31),(10,25,49,32),(11,26,50,33),(12,27,51,34),(13,28,52,35),(14,15,53,36)]])

140 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F7G···7AA14A···14R14S···14CC28A···28X
order122244447···77···714···1414···1428···28
size111177771···12···21···12···27···7

140 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C4C7C14C14C28D7Dic7D14C7xD7C7xDic7D7xC14
kernelC14xDic7C7xDic7C142C7xC14C2xDic7Dic7C2xC14C14C2xC14C14C14C22C2C2
# reps1214612624363183618

Matrix representation of C14xDic7 in GL3(F29) generated by

2000
0220
0022
,
2800
0230
0024
,
1700
0028
0280
G:=sub<GL(3,GF(29))| [20,0,0,0,22,0,0,0,22],[28,0,0,0,23,0,0,0,24],[17,0,0,0,0,28,0,28,0] >;

C14xDic7 in GAP, Magma, Sage, TeX

C_{14}\times {\rm Dic}_7
% in TeX

G:=Group("C14xDic7");
// GroupNames label

G:=SmallGroup(392,26);
// by ID

G=gap.SmallGroup(392,26);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,140,8404]);
// Polycyclic

G:=Group<a,b,c|a^14=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C14xDic7 in TeX

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