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G = D18.4D6order 432 = 24·33

4th non-split extension by D18 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D18.4D6, D6.4D18, Dic9.6D6, C62.67D6, Dic3.6D18, C9:D4:1S3, C3:D4:1D9, (S3xC6).6D6, (C2xC18).5D6, (C2xC6).5D18, D6:D9:4C2, (Dic3xD9):2C2, (S3xDic9):4C2, C9:4(D4:2S3), C3:4(D4:2D9), C22.3(S3xD9), C9:Dic6:6C2, (C3xDic3).6D6, (C6xD9).4C22, C6.22(C22xD9), (S3xC18).4C22, (C3xC18).22C23, (C6xC18).16C22, C18.22(C22xS3), C3.1(D6.4D6), C9:Dic3.12C22, (C9xDic3).6C22, (C3xDic9).6C22, C32.5(D4:2S3), (C2xC6).5S32, C6.41(C2xS32), C2.22(C2xS3xD9), (C9xC3:D4):2C2, (C3xC9:D4):2C2, (C3xC9):10(C4oD4), (C2xC9:Dic3):7C2, (C3xC3:D4).2S3, (C3xC6).90(C22xS3), SmallGroup(432,310)

Series: Derived Chief Lower central Upper central

C1C3xC18 — D18.4D6
C1C3C32C3xC9C3xC18S3xC18S3xDic9 — D18.4D6
C3xC9C3xC18 — D18.4D6
C1C2C22

Generators and relations for D18.4D6
 G = < a,b,c,d | a18=b2=1, c6=d2=a9, bab=cac-1=dad-1=a-1, cbc-1=a7b, dbd-1=a16b, dcd-1=c5 >

Subgroups: 728 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, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3:D4, C3xD4, C3xC9, Dic9, Dic9, C36, D18, C2xC18, C2xC18, C3xDic3, C3xDic3, C3:Dic3, S3xC6, S3xC6, C62, D4:2S3, C3xD9, S3xC9, C3xC18, C3xC18, Dic18, C4xD9, C2xDic9, C9:D4, C9:D4, D4xC9, S3xDic3, D6:S3, C32:2Q8, C3xC3:D4, C3xC3:D4, C2xC3:Dic3, C3xDic9, C9xDic3, C9:Dic3, C6xD9, S3xC18, C6xC18, D4:2D9, D6.4D6, C9:Dic6, Dic3xD9, S3xDic9, D6:D9, C3xC9:D4, C9xC3:D4, C2xC9:Dic3, D18.4D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, D9, C22xS3, D18, S32, D4:2S3, C22xD9, C2xS32, S3xD9, D4:2D9, D6.4D6, C2xS3xD9, D18.4D6

Smallest permutation representation of D18.4D6
On 72 points
Generators in S72
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(37 56)(38 55)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)
(1 34 16 19 13 22 10 25 7 28 4 31)(2 33 17 36 14 21 11 24 8 27 5 30)(3 32 18 35 15 20 12 23 9 26 6 29)(37 69 40 66 43 63 46 60 49 57 52 72)(38 68 41 65 44 62 47 59 50 56 53 71)(39 67 42 64 45 61 48 58 51 55 54 70)
(1 63 10 72)(2 62 11 71)(3 61 12 70)(4 60 13 69)(5 59 14 68)(6 58 15 67)(7 57 16 66)(8 56 17 65)(9 55 18 64)(19 49 28 40)(20 48 29 39)(21 47 30 38)(22 46 31 37)(23 45 32 54)(24 44 33 53)(25 43 34 52)(26 42 35 51)(27 41 36 50)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C···6G6H6I9A9B9C9D9E9F12A12B18A18B18C18D···18O18P18Q18R36A36B36C
order1222233344444666···666999999121218181818···18181818363636
size112618224618272754224···4123622244412362224···4121212121212

51 irreducible representations

dim111111112222222222222444444444
type+++++++++++++++++++++--++--+-
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D6D6D6C4oD4D9D18D18D18S32D4:2S3D4:2S3C2xS32S3xD9D4:2D9D6.4D6C2xS3xD9D18.4D6
kernelD18.4D6C9:Dic6Dic3xD9S3xDic9D6:D9C3xC9:D4C9xC3:D4C2xC9:Dic3C9:D4C3xC3:D4Dic9D18C2xC18C3xDic3S3xC6C62C3xC9C3:D4Dic3D6C2xC6C2xC6C9C32C6C22C3C3C2C1
# reps111111111111111123333111133236

Matrix representation of D18.4D6 in GL6(F37)

100000
010000
0016000
000700
0000360
0000036
,
100000
010000
000700
0016000
0000036
0000360
,
1360000
100000
000100
001000
000001
0000360
,
100000
1360000
000100
001000
0000310
0000031

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,7,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0],[1,1,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0],[1,1,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,0,0,0,0,0,0,31] >;

D18.4D6 in GAP, Magma, Sage, TeX

D_{18}._4D_6
% in TeX

G:=Group("D18.4D6");
// GroupNames label

G:=SmallGroup(432,310);
// by ID

G=gap.SmallGroup(432,310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^2=1,c^6=d^2=a^9,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^7*b,d*b*d^-1=a^16*b,d*c*d^-1=c^5>;
// generators/relations

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