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G = C9×C7⋊C3order 189 = 33·7

Direct product of C9 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C9×C7⋊C3, C631C3, C21.1C32, C7⋊C94C3, C71(C3×C9), C3.1(C3×C7⋊C3), (C3×C7⋊C3).3C3, SmallGroup(189,3)

Series: Derived Chief Lower central Upper central

C1C7 — C9×C7⋊C3
C1C7C21C3×C7⋊C3 — C9×C7⋊C3
C7 — C9×C7⋊C3
C1C9

Generators and relations for C9×C7⋊C3
 G = < a,b,c | a9=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C3
7C3
7C9
7C32
7C9
7C3×C9

Smallest permutation representation of C9×C7⋊C3
On 63 points
Generators in S63
(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)
(1 46 29 21 39 16 56)(2 47 30 22 40 17 57)(3 48 31 23 41 18 58)(4 49 32 24 42 10 59)(5 50 33 25 43 11 60)(6 51 34 26 44 12 61)(7 52 35 27 45 13 62)(8 53 36 19 37 14 63)(9 54 28 20 38 15 55)
(1 7 4)(2 8 5)(3 9 6)(10 21 62)(11 22 63)(12 23 55)(13 24 56)(14 25 57)(15 26 58)(16 27 59)(17 19 60)(18 20 61)(28 44 48)(29 45 49)(30 37 50)(31 38 51)(32 39 52)(33 40 53)(34 41 54)(35 42 46)(36 43 47)

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

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

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

C9×C7⋊C3 is a maximal subgroup of   C95F7

45 conjugacy classes

class 1 3A3B3C···3H7A7B9A···9F9G···9R21A21B21C21D63A···63L
order1333···3779···99···92121212163···63
size1117···7331···17···733333···3

45 irreducible representations

dim11111333
type+
imageC1C3C3C3C9C7⋊C3C3×C7⋊C3C9×C7⋊C3
kernelC9×C7⋊C3C7⋊C9C63C3×C7⋊C3C7⋊C3C9C3C1
# reps1422182412

Matrix representation of C9×C7⋊C3 in GL3(𝔽127) generated by

2200
0220
0022
,
1041051
100
010
,
1900
37108108
0190
G:=sub<GL(3,GF(127))| [22,0,0,0,22,0,0,0,22],[104,1,0,105,0,1,1,0,0],[19,37,0,0,108,19,0,108,0] >;

C9×C7⋊C3 in GAP, Magma, Sage, TeX

C_9\times C_7\rtimes C_3
% in TeX

G:=Group("C9xC7:C3");
// GroupNames label

G:=SmallGroup(189,3);
// by ID

G=gap.SmallGroup(189,3);
# by ID

G:=PCGroup([4,-3,-3,-3,-7,29,867]);
// Polycyclic

G:=Group<a,b,c|a^9=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C9×C7⋊C3 in TeX

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