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G = C9×D12  order 216 = 23·33

Direct product of C9 and D12

direct product, metacyclic, supersoluble, monomial

Aliases: C9×D12, C36⋊3S3, C12⋊1C18, D6⋊1C18, C18.21D6, C4⋊(S3×C9), (C3×C9)⋊4D4, C3⋊1(D4×C9), (C3×C36)⋊6C2, (C3×D12).C3, (S3×C18)⋊1C2, (S3×C6).1C6, C2.4(S3×C18), C6.31(S3×C6), C6.3(C2×C18), C3.4(C3×D12), C12.16(C3×S3), (C3×C12).11C6, C32.2(C3×D4), (C3×C18).10C22, (C3×C6).20(C2×C6), SmallGroup(216,48)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C9×D12
Chief series
C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — C9×D12
Lower central
C3 — C6 — C9×D12
Upper central
C1 — C18 — C36

Generators and relations for C9×D12
 G = < a,b,c | a9=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
C2
6C2
6C2
C3
C3
2C3
C4
3C22
3C22
C6
C6
2S3
2C6
2S3
6C6
6C6
C32
C9
2C9
3D4
D6
C12
C12
D6
2C12
3C2×C6
3C2×C6
C3×C6
C18
2C18
2C3×S3
2C3×S3
6C18
6C18
C3×C9
D12
3C3×D4
C3×C12
C36
S3×C6
S3×C6
2C36
3C2×C18
3C2×C18
C3×C18
2S3×C9
2S3×C9
C3×D12
3D4×C9
S3×C18
C3×C36
S3×C18
C9×D12

Smallest permutation representation of C9×D12
►On 72 points
Generators in S72
(1 55 62 5 59 66 9 51 70)(2 56 63 6 60 67 10 52 71)(3 57 64 7 49 68 11 53 72)(4 58 65 8 50 69 12 54 61)(13 35 40 21 31 48 17 27 44)(14 36 41 22 32 37 18 28 45)(15 25 42 23 33 38 19 29 46)(16 26 43 24 34 39 20 30 47)

(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 64 65 66 67 68 69 70 71 72)

(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 72)(22 71)(23 70)(24 69)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)
 
G:=sub<Sym(72)| (1,55,62,5,59,66,9,51,70)(2,56,63,6,60,67,10,52,71)(3,57,64,7,49,68,11,53,72)(4,58,65,8,50,69,12,54,61)(13,35,40,21,31,48,17,27,44)(14,36,41,22,32,37,18,28,45)(15,25,42,23,33,38,19,29,46)(16,26,43,24,34,39,20,30,47), (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)(61,62,63,64,65,66,67,68,69,70,71,72), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)>;
 
G:=Group( (1,55,62,5,59,66,9,51,70)(2,56,63,6,60,67,10,52,71)(3,57,64,7,49,68,11,53,72)(4,58,65,8,50,69,12,54,61)(13,35,40,21,31,48,17,27,44)(14,36,41,22,32,37,18,28,45)(15,25,42,23,33,38,19,29,46)(16,26,43,24,34,39,20,30,47), (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)(61,62,63,64,65,66,67,68,69,70,71,72), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57) );
 
G=PermutationGroup([[(1,55,62,5,59,66,9,51,70),(2,56,63,6,60,67,10,52,71),(3,57,64,7,49,68,11,53,72),(4,58,65,8,50,69,12,54,61),(13,35,40,21,31,48,17,27,44),(14,36,41,22,32,37,18,28,45),(15,25,42,23,33,38,19,29,46),(16,26,43,24,34,39,20,30,47)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,72),(22,71),(23,70),(24,69),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57)]])
 

C9×D12 is a maximal subgroup of
 D36⋊S3  C9⋊D24  D12.D9  C36.D6  D12⋊5D9  D12⋊D9  C36⋊D6  S3×D4×C9

81 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I9A···9F9G···9L12A···12H18A···18F18G···18L18M···18X36A···36R
order12223333346666666669···99···912···1218···1818···1818···1836···36
size11661122221122266661···12···22···21···12···26···62···2

81 irreducible representations

dim111111111222222222222
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D4D6C3×S3D12C3×D4S3×C6S3×C9D4×C9C3×D12S3×C18C9×D12
kernelC9×D12C3×C36S3×C18C3×D12C3×C12S3×C6D12C12D6C36C3×C9C18C12C9C32C6C4C3C3C2C1
# reps11222466121112222664612

Matrix representation of C9×D12 ►in GL2(𝔽37) generated by

70
07
,
230
029
,
030
210
G:=sub<GL(2,GF(37))| [7,0,0,7],[23,0,0,29],[0,21,30,0] >;
 

C9×D12 in GAP, Magma, Sage, TeX 

C_9\times D_{12}
 
% in TeX
 
G:=Group("C9xD12");
 
// GroupNames label
 
G:=SmallGroup(216,48);
 
// by ID
 
G=gap.SmallGroup(216,48);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,122,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C9×D12 in TeX

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