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G = C7×C7⋊C9order 441 = 32·72

Direct product of C7 and C7⋊C9

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

Aliases: C7×C7⋊C9, C7⋊C63, C21.C21, C721C9, C21.5(C7⋊C3), (C7×C21).1C3, C3.(C7×C7⋊C3), SmallGroup(441,5)

Series: Derived Chief Lower central Upper central

C1C7 — C7×C7⋊C9
C1C7C21C7×C21 — C7×C7⋊C9
C7 — C7×C7⋊C9
C1C21

Generators and relations for C7×C7⋊C9
 G = < a,b,c | a7=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

3C7
3C7
7C9
3C21
3C21
7C63

Smallest permutation representation of C7×C7⋊C9
On 63 points
Generators in S63
(1 21 17 39 32 51 56)(2 22 18 40 33 52 57)(3 23 10 41 34 53 58)(4 24 11 42 35 54 59)(5 25 12 43 36 46 60)(6 26 13 44 28 47 61)(7 27 14 45 29 48 62)(8 19 15 37 30 49 63)(9 20 16 38 31 50 55)
(1 21 17 39 32 51 56)(2 33 22 52 18 57 40)(3 10 34 58 23 41 53)(4 24 11 42 35 54 59)(5 36 25 46 12 60 43)(6 13 28 61 26 44 47)(7 27 14 45 29 48 62)(8 30 19 49 15 63 37)(9 16 31 55 20 38 50)
(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 57 58 59 60 61 62 63)

G:=sub<Sym(63)| (1,21,17,39,32,51,56)(2,22,18,40,33,52,57)(3,23,10,41,34,53,58)(4,24,11,42,35,54,59)(5,25,12,43,36,46,60)(6,26,13,44,28,47,61)(7,27,14,45,29,48,62)(8,19,15,37,30,49,63)(9,20,16,38,31,50,55), (1,21,17,39,32,51,56)(2,33,22,52,18,57,40)(3,10,34,58,23,41,53)(4,24,11,42,35,54,59)(5,36,25,46,12,60,43)(6,13,28,61,26,44,47)(7,27,14,45,29,48,62)(8,30,19,49,15,63,37)(9,16,31,55,20,38,50), (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)(55,56,57,58,59,60,61,62,63)>;

G:=Group( (1,21,17,39,32,51,56)(2,22,18,40,33,52,57)(3,23,10,41,34,53,58)(4,24,11,42,35,54,59)(5,25,12,43,36,46,60)(6,26,13,44,28,47,61)(7,27,14,45,29,48,62)(8,19,15,37,30,49,63)(9,20,16,38,31,50,55), (1,21,17,39,32,51,56)(2,33,22,52,18,57,40)(3,10,34,58,23,41,53)(4,24,11,42,35,54,59)(5,36,25,46,12,60,43)(6,13,28,61,26,44,47)(7,27,14,45,29,48,62)(8,30,19,49,15,63,37)(9,16,31,55,20,38,50), (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)(55,56,57,58,59,60,61,62,63) );

G=PermutationGroup([[(1,21,17,39,32,51,56),(2,22,18,40,33,52,57),(3,23,10,41,34,53,58),(4,24,11,42,35,54,59),(5,25,12,43,36,46,60),(6,26,13,44,28,47,61),(7,27,14,45,29,48,62),(8,19,15,37,30,49,63),(9,20,16,38,31,50,55)], [(1,21,17,39,32,51,56),(2,33,22,52,18,57,40),(3,10,34,58,23,41,53),(4,24,11,42,35,54,59),(5,36,25,46,12,60,43),(6,13,28,61,26,44,47),(7,27,14,45,29,48,62),(8,30,19,49,15,63,37),(9,16,31,55,20,38,50)], [(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),(55,56,57,58,59,60,61,62,63)]])

105 conjugacy classes

class 1 3A3B7A···7F7G···7T9A···9F21A···21L21M···21AN63A···63AJ
order1337···77···79···921···2121···2163···63
size1111···13···37···71···13···37···7

105 irreducible representations

dim1111113333
type+
imageC1C3C7C9C21C63C7⋊C3C7⋊C9C7×C7⋊C3C7×C7⋊C9
kernelC7×C7⋊C9C7×C21C7⋊C9C72C21C7C21C7C3C1
# reps12661236241224

Matrix representation of C7×C7⋊C9 in GL3(𝔽127) generated by

1600
0160
0016
,
1600
020
004
,
010
001
1900
G:=sub<GL(3,GF(127))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,2,0,0,0,4],[0,0,19,1,0,0,0,1,0] >;

C7×C7⋊C9 in GAP, Magma, Sage, TeX

C_7\times C_7\rtimes C_9
% in TeX

G:=Group("C7xC7:C9");
// GroupNames label

G:=SmallGroup(441,5);
// by ID

G=gap.SmallGroup(441,5);
# by ID

G:=PCGroup([4,-3,-7,-3,-7,84,2019]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C7×C7⋊C9 in TeX

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