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G = Dic3.D18order 432 = 24·33

5th non-split extension by Dic3 of D18 acting via D18/D9=C2

metabelian, supersoluble, monomial

Aliases: D6.3D18, C62.66D6, Dic9.12D6, Dic3.5D18, C3:D4:3D9, (S3xC6).5D6, (C2xC18).4D6, C9:D12:4C2, C9:4(C4oD12), (C6xDic9):7C2, (S3xDic9):3C2, (C2xDic9):3S3, (C2xC6).20D18, C3:3(D4:2D9), C22.1(S3xD9), C9:Dic6:5C2, C18.D6:2C2, (C3xDic3).5D6, C6.21(C22xD9), C6.D18:2C2, (S3xC18).3C22, C18.21(C22xS3), (C3xC18).21C23, (C6xC18).15C22, C9:Dic3.6C22, C3.2(D6.3D6), (C9xDic3).5C22, C32.4(D4:2S3), (C3xDic9).12C22, (C2xC6).4S32, C6.40(C2xS32), C2.21(C2xS3xD9), (C3xC9):9(C4oD4), (C9xC3:D4):1C2, (C3xC3:D4).1S3, (C2xC9:S3).4C22, (C3xC6).89(C22xS3), SmallGroup(432,309)

Series: Derived Chief Lower central Upper central

C1C3xC18 — Dic3.D18
C1C3C32C3xC9C3xC18S3xC18S3xDic9 — Dic3.D18
C3xC9C3xC18 — Dic3.D18
C1C2C22

Generators and relations for Dic3.D18
 G = < a,b,c,d | a6=c18=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 828 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C4oD4, D9, C18, C18, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C3xD4, C3xC9, Dic9, Dic9, C36, D18, C2xC18, C2xC18, C3xDic3, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C62, C4oD12, D4:2S3, S3xC9, C9:S3, C3xC18, C3xC18, Dic18, C4xD9, C2xDic9, C2xDic9, C9:D4, D4xC9, S3xDic3, C6.D6, C3:D12, C32:2Q8, C6xDic3, C3xC3:D4, C32:7D4, C3xDic9, C9xDic3, C9:Dic3, S3xC18, C2xC9:S3, C6xC18, D4:2D9, D6.3D6, C9:Dic6, C18.D6, S3xDic9, C9:D12, C6xDic9, C9xC3:D4, C6.D18, Dic3.D18
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, D9, C22xS3, D18, S32, C4oD12, D4:2S3, C22xD9, C2xS32, S3xD9, D4:2D9, D6.3D6, C2xS3xD9, Dic3.D18

Smallest permutation representation of Dic3.D18
On 72 points
Generators in S72
(1 34 7 22 13 28)(2 29 14 23 8 35)(3 36 9 24 15 30)(4 31 16 25 10 19)(5 20 11 26 17 32)(6 33 18 27 12 21)(37 64 49 58 43 70)(38 71 44 59 50 65)(39 66 51 60 45 72)(40 55 46 61 52 67)(41 68 53 62 47 56)(42 57 48 63 54 69)
(1 31 22 10)(2 11 23 32)(3 33 24 12)(4 13 25 34)(5 35 26 14)(6 15 27 36)(7 19 28 16)(8 17 29 20)(9 21 30 18)(37 67 58 46)(38 47 59 68)(39 69 60 48)(40 49 61 70)(41 71 62 50)(42 51 63 72)(43 55 64 52)(44 53 65 56)(45 57 66 54)
(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 63 22 42)(2 62 23 41)(3 61 24 40)(4 60 25 39)(5 59 26 38)(6 58 27 37)(7 57 28 54)(8 56 29 53)(9 55 30 52)(10 72 31 51)(11 71 32 50)(12 70 33 49)(13 69 34 48)(14 68 35 47)(15 67 36 46)(16 66 19 45)(17 65 20 44)(18 64 21 43)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I9A9B9C9D9E9F12A12B12C12D12E18A18B18C18D···18O18P18Q18R36A36B36C
order1222233344444666666666999999121212121218181818···18181818363636
size1126542246991854222244441222244412181818182224···4121212121212

54 irreducible representations

dim11111111222222222222244444444
type++++++++++++++++++++-++-+
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D6D6C4oD4D9D18D18D18C4oD12S32D4:2S3C2xS32S3xD9D4:2D9D6.3D6C2xS3xD9Dic3.D18
kernelDic3.D18C9:Dic6C18.D6S3xDic9C9:D12C6xDic9C9xC3:D4C6.D18C2xDic9C3xC3:D4Dic9C2xC18C3xDic3S3xC6C62C3xC9C3:D4Dic3D6C2xC6C9C2xC6C32C6C22C3C3C2C1
# reps11111111112111123333411133236

Matrix representation of Dic3.D18 in GL4(F37) generated by

27000
331100
0010
0001
,
29500
24800
0010
0001
,
29500
17800
00120
00034
,
263000
281100
00034
00120
G:=sub<GL(4,GF(37))| [27,33,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[29,24,0,0,5,8,0,0,0,0,1,0,0,0,0,1],[29,17,0,0,5,8,0,0,0,0,12,0,0,0,0,34],[26,28,0,0,30,11,0,0,0,0,0,12,0,0,34,0] >;

Dic3.D18 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_{18}
% in TeX

G:=Group("Dic3.D18");
// GroupNames label

G:=SmallGroup(432,309);
// by ID

G=gap.SmallGroup(432,309);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,3091,662,4037,7069]);
// Polycyclic

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

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