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G = C7×D9order 126 = 2·32·7

Direct product of C7 and D9

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

Aliases: C7×D9, C9⋊C14, C632C2, C21.2S3, C3.(S3×C7), SmallGroup(126,3)

Series: Derived Chief Lower central Upper central

C1C9 — C7×D9
C1C3C9C63 — C7×D9
C9 — C7×D9
C1C7

Generators and relations for C7×D9
 G = < a,b,c | a7=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C14
3S3×C7

Smallest permutation representation of C7×D9
On 63 points
Generators in S63
(1 56 47 38 29 20 11)(2 57 48 39 30 21 12)(3 58 49 40 31 22 13)(4 59 50 41 32 23 14)(5 60 51 42 33 24 15)(6 61 52 43 34 25 16)(7 62 53 44 35 26 17)(8 63 54 45 36 27 18)(9 55 46 37 28 19 10)
(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 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 20)(21 27)(22 26)(23 25)(28 29)(30 36)(31 35)(32 34)(37 38)(39 45)(40 44)(41 43)(46 47)(48 54)(49 53)(50 52)(55 56)(57 63)(58 62)(59 61)

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

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

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

C7×D9 is a maximal subgroup of   C63⋊C6  C636C6

42 conjugacy classes

class 1  2  3 7A···7F9A9B9C14A···14F21A···21F63A···63R
order1237···799914···1421···2163···63
size1921···12229···92···22···2

42 irreducible representations

dim11112222
type++++
imageC1C2C7C14S3D9S3×C7C7×D9
kernelC7×D9C63D9C9C21C7C3C1
# reps116613618

Matrix representation of C7×D9 in GL2(𝔽127) generated by

320
032
,
969
118105
,
10531
922
G:=sub<GL(2,GF(127))| [32,0,0,32],[96,118,9,105],[105,9,31,22] >;

C7×D9 in GAP, Magma, Sage, TeX

C_7\times D_9
% in TeX

G:=Group("C7xD9");
// GroupNames label

G:=SmallGroup(126,3);
// by ID

G=gap.SmallGroup(126,3);
# by ID

G:=PCGroup([4,-2,-7,-3,-3,842,82,1347]);
// Polycyclic

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

Export

Subgroup lattice of C7×D9 in TeX

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