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G = C7×C7⋊C8order 392 = 23·72

Direct product of C7 and C7⋊C8

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

Aliases: C7×C7⋊C8, C7⋊C56, C14.C28, C723C8, C28.8D7, C28.2C14, C14.4Dic7, C4.2(C7×D7), C2.(C7×Dic7), (C7×C28).3C2, (C7×C14).2C4, SmallGroup(392,14)

Series: Derived Chief Lower central Upper central

C1C7 — C7×C7⋊C8
C1C7C14C28C7×C28 — C7×C7⋊C8
C7 — C7×C7⋊C8
C1C28

Generators and relations for C7×C7⋊C8
 G = < a,b,c | a7=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C7
2C7
2C7
2C14
2C14
2C14
7C8
2C28
2C28
2C28
7C56

Smallest permutation representation of C7×C7⋊C8
On 56 points
Generators in S56
(1 34 19 49 45 30 10)(2 35 20 50 46 31 11)(3 36 21 51 47 32 12)(4 37 22 52 48 25 13)(5 38 23 53 41 26 14)(6 39 24 54 42 27 15)(7 40 17 55 43 28 16)(8 33 18 56 44 29 9)
(1 45 34 30 19 10 49)(2 50 11 20 31 35 46)(3 47 36 32 21 12 51)(4 52 13 22 25 37 48)(5 41 38 26 23 14 53)(6 54 15 24 27 39 42)(7 43 40 28 17 16 55)(8 56 9 18 29 33 44)
(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)

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

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

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

140 conjugacy classes

class 1  2 4A4B7A···7F7G···7AA8A8B8C8D14A···14F14G···14AA28A···28L28M···28BB56A···56X
order12447···77···7888814···1414···1428···2828···2856···56
size11111···12···277771···12···21···12···27···7

140 irreducible representations

dim11111111222222
type+++-
imageC1C2C4C7C8C14C28C56D7Dic7C7⋊C8C7×D7C7×Dic7C7×C7⋊C8
kernelC7×C7⋊C8C7×C28C7×C14C7⋊C8C72C28C14C7C28C14C7C4C2C1
# reps1126461224336181836

Matrix representation of C7×C7⋊C8 in GL2(𝔽29) generated by

230
023
,
70
025
,
017
10
G:=sub<GL(2,GF(29))| [23,0,0,23],[7,0,0,25],[0,1,17,0] >;

C7×C7⋊C8 in GAP, Magma, Sage, TeX

C_7\times C_7\rtimes C_8
% in TeX

G:=Group("C7xC7:C8");
// GroupNames label

G:=SmallGroup(392,14);
// by ID

G=gap.SmallGroup(392,14);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-7,70,42,8404]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×C7⋊C8 in TeX

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