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G = C18.D12order 432 = 24·33

3rd non-split extension by C18 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: Dic61D9, C36.14D6, C18.14D12, C12.32D18, C9⋊C83S3, C12.6S32, C4.3(S3×D9), (C3×C9)⋊6SD16, C92(C24⋊C2), (C9×Dic6)⋊2C2, (C3×C18).10D4, (C3×C12).78D6, C6.3(C9⋊D4), C36⋊S3.3C2, C31(Q82D9), (C3×Dic6).2S3, C2.6(C9⋊D12), (C3×C36).13C22, C6.16(C3⋊D12), C3.2(C325SD16), C32.3(Q82S3), (C3×C9⋊C8)⋊3C2, (C3×C6).46(C3⋊D4), SmallGroup(432,73)

Series: Derived Chief Lower central Upper central

C1C3×C36 — C18.D12
C1C3C32C3×C9C3×C18C3×C36C9×Dic6 — C18.D12
C3×C9C3×C18C3×C36 — C18.D12
C1C2C4

Generators and relations for C18.D12
 G = < a,b,c | a12=c2=1, b18=a6, bab-1=cac=a-1, cbc=a3b17 >

Subgroups: 676 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C3×C9, C36, C36, D18, C3×Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, Q82S3, C9⋊S3, C3×C18, C9⋊C8, D36, Q8×C9, C3×C3⋊C8, C3×Dic6, C12⋊S3, C9×Dic3, C3×C36, C2×C9⋊S3, Q82D9, C325SD16, C3×C9⋊C8, C9×Dic6, C36⋊S3, C18.D12
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, Q82S3, C9⋊D4, C3⋊D12, S3×D9, Q82D9, C325SD16, C9⋊D12, C18.D12

Smallest permutation representation of C18.D12
On 72 points
Generators in S72
(1 40 7 46 13 52 19 58 25 64 31 70)(2 71 32 65 26 59 20 53 14 47 8 41)(3 42 9 48 15 54 21 60 27 66 33 72)(4 37 34 67 28 61 22 55 16 49 10 43)(5 44 11 50 17 56 23 62 29 68 35 38)(6 39 36 69 30 63 24 57 18 51 12 45)
(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 72)(2 26)(3 70)(4 24)(5 68)(6 22)(7 66)(8 20)(9 64)(10 18)(11 62)(12 16)(13 60)(15 58)(17 56)(19 54)(21 52)(23 50)(25 48)(27 46)(28 36)(29 44)(30 34)(31 42)(33 40)(35 38)(37 63)(39 61)(41 59)(43 57)(45 55)(47 53)(49 51)(65 71)(67 69)

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

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

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

51 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C8A8B9A9B9C9D9E9F12A12B12C12D12E12F12G18A18B18C18D18E18F24A24B24C24D36A···36I36J···36O
order1223334466688999999121212121212121818181818182424242436···3636···36
size111082242122241818222444224441212222444181818184···412···12

51 irreducible representations

dim111122222222222244444444
type++++++++++++++++++++
imageC1C2C2C2S3S3D4D6D6SD16D9D12C3⋊D4D18C24⋊C2C9⋊D4S32Q82S3C3⋊D12S3×D9Q82D9C325SD16C9⋊D12C18.D12
kernelC18.D12C3×C9⋊C8C9×Dic6C36⋊S3C9⋊C8C3×Dic6C3×C18C36C3×C12C3×C9Dic6C18C3×C6C12C9C6C12C32C6C4C3C3C2C1
# reps111111111232234611133236

Matrix representation of C18.D12 in GL6(𝔽73)

72710000
110000
00727200
001000
000010
000001
,
61610000
6120000
0072000
001100
00002842
00003170
,
120000
0720000
001000
00727200
00004531
0000328

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,71,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[61,6,0,0,0,0,61,12,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,28,31,0,0,0,0,42,70],[1,0,0,0,0,0,2,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,45,3,0,0,0,0,31,28] >;

C18.D12 in GAP, Magma, Sage, TeX

C_{18}.D_{12}
% in TeX

G:=Group("C18.D12");
// GroupNames label

G:=SmallGroup(432,73);
// by ID

G=gap.SmallGroup(432,73);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c|a^12=c^2=1,b^18=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^17>;
// generators/relations

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