Copied to
clipboard

G = C2×C9⋊C12  order 216 = 23·33

Direct product of C2 and C9⋊C12

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C9⋊C12, C18⋊C12, Dic9⋊3C6, C62.2S3, (C2×C18).C6, C9⋊2(C2×C12), (C2×Dic9)⋊C3, C6.19(S3×C6), C18.4(C2×C6), C22.(C9⋊C6), (C3×C6).14D6, (C3×C6).4Dic3, C3.3(C6×Dic3), C6.6(C3×Dic3), C32.(C2×Dic3), (C2×3- 1+2)⋊C4, 3- 1+2⋊2(C2×C4), (C22×3- 1+2).C2, (C2×3- 1+2).4C22, C2.2(C2×C9⋊C6), (C2×C6).14(C3×S3), SmallGroup(216,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2×C9⋊C12
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C9⋊C12 — C2×C9⋊C12
Lower central
C9 — C2×C9⋊C12
Upper central
C1 — C22

Generators and relations for C2×C9⋊C12
 G = < a,b,c | a2=b9=c12=1, ab=ba, ac=ca, cbc-1=b5 >

C1
C2
C2
C2
C3
3C3
C22
9C4
9C4
C6
C6
C6
3C6
3C6
3C6
C32
C9
2C9
9C2×C4
C2×C6
3C2×C6
3Dic3
3Dic3
9C12
9C12
C18
C3×C6
C18
C3×C6
C18
C3×C6
2C18
2C18
2C18
3- 1+2
3C2×Dic3
9C2×C12
C62
C2×C18
Dic9
Dic9
2C2×C18
3C3×Dic3
3C3×Dic3
C2×3- 1+2
C2×3- 1+2
C2×3- 1+2
C2×Dic9
3C6×Dic3
C9⋊C12
C9⋊C12
C22×3- 1+2
C2×C9⋊C12

Smallest permutation representation of C2×C9⋊C12
►On 72 points
Generators in S72
(1 12)(2 9)(3 10)(4 11)(5 23)(6 24)(7 21)(8 22)(13 17)(14 18)(15 19)(16 20)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)

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

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

C2×C9⋊C12 is a maximal subgroup of   Dic9⋊C12  C36⋊C12  D18⋊C12  C62.27D6  C2×C4×C9⋊C6  Dic18⋊2C6
C2×C9⋊C12 is a maximal quotient of   C36.C12  C36⋊C12  C62.27D6

40 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C6D···6I9A9B9C12A···12H18A···18I
order122233344446666···699912···1218···18
size111123399992223···36669···96···6

40 irreducible representations

dim11111111222222666
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6C9⋊C6C9⋊C12C2×C9⋊C6
kernelC2×C9⋊C12C9⋊C12C22×3- 1+2C2×Dic9C2×3- 1+2Dic9C2×C18C18C62C3×C6C3×C6C2×C6C6C6C22C2C2
# reps12124428121242121

Matrix representation of C2×C9⋊C12 ►in GL8(𝔽37)

10000000
01000000
003600000
000360000
000036000
000003600
000000360
000000036
,
361000000
360000000
000100000
000010000
00100000
00000001
000002600
000000260
,
298000000
08000000
00000100
000000260
000000010
003600000
000110000
000027000

G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0],[29,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,10,0,0,0] >;
 

C2×C9⋊C12 in GAP, Magma, Sage, TeX 

C_2\times C_9\rtimes C_{12}
 
% in TeX
 
G:=Group("C2xC9:C12");
 
// GroupNames label
 
G:=SmallGroup(216,61);
 
// by ID
 
G=gap.SmallGroup(216,61);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,3604,736,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^9=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C9⋊C12 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁