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G = C9×D7order 126 = 2·32·7

Direct product of C9 and D7

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9×D7, C633C2, C73C18, C21.3C6, C3.(C3×D7), (C3×D7).2C3, SmallGroup(126,4)

Series: Derived Chief Lower central Upper central

C1C7 — C9×D7
C1C7C21C63 — C9×D7
C7 — C9×D7
C1C9

Generators and relations for C9×D7
 G = < a,b,c | a9=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C6
7C18

Smallest permutation representation of C9×D7
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 56 44 32 48 17 20)(2 57 45 33 49 18 21)(3 58 37 34 50 10 22)(4 59 38 35 51 11 23)(5 60 39 36 52 12 24)(6 61 40 28 53 13 25)(7 62 41 29 54 14 26)(8 63 42 30 46 15 27)(9 55 43 31 47 16 19)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 55)(17 56)(18 57)(37 50)(38 51)(39 52)(40 53)(41 54)(42 46)(43 47)(44 48)(45 49)

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

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

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

C9×D7 is a maximal subgroup of   C7⋊C54  C93F7  C94F7

45 conjugacy classes

class 1  2 3A3B6A6B7A7B7C9A···9F18A···18F21A···21F63A···63R
order1233667779···918···1821···2163···63
size1711772221···17···72···22···2

45 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D7C3×D7C9×D7
kernelC9×D7C63C3×D7C21D7C7C9C3C1
# reps1122663618

Matrix representation of C9×D7 in GL2(𝔽127) generated by

1030
0103
,
1261
8937
,
1260
891
G:=sub<GL(2,GF(127))| [103,0,0,103],[126,89,1,37],[126,89,0,1] >;

C9×D7 in GAP, Magma, Sage, TeX

C_9\times D_7
% in TeX

G:=Group("C9xD7");
// GroupNames label

G:=SmallGroup(126,4);
// by ID

G=gap.SmallGroup(126,4);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,29,1731]);
// Polycyclic

G:=Group<a,b,c|a^9=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D7 in TeX

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