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## G = Dic3.D18order 432 = 24·33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Dic3.D18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — S3×Dic9 — Dic3.D18
 Lower central C3×C9 — C3×C18 — Dic3.D18
 Upper central C1 — C2 — C22

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, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×C9, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×C9, C9⋊S3, C3×C18, C3×C18, Dic18, C4×D9, C2×Dic9, C2×Dic9, C9⋊D4, D4×C9, S3×Dic3, C6.D6, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C327D4, C3×Dic9, C9×Dic3, C9⋊Dic3, S3×C18, C2×C9⋊S3, C6×C18, D42D9, D6.3D6, C9⋊Dic6, C18.D6, S3×Dic9, C9⋊D12, C6×Dic9, C9×C3⋊D4, C6.D18, Dic3.D18
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, D42S3, C22×D9, C2×S32, S3×D9, D42D9, D6.3D6, C2×S3×D9, 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 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18O 18P 18Q 18R 36A 36B 36C order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 18 18 18 36 36 36 size 1 1 2 6 54 2 2 4 6 9 9 18 54 2 2 2 2 4 4 4 4 12 2 2 2 4 4 4 12 18 18 18 18 2 2 2 4 ··· 4 12 12 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 D9 D18 D18 D18 C4○D12 S32 D4⋊2S3 C2×S32 S3×D9 D4⋊2D9 D6.3D6 C2×S3×D9 Dic3.D18 kernel Dic3.D18 C9⋊Dic6 C18.D6 S3×Dic9 C9⋊D12 C6×Dic9 C9×C3⋊D4 C6.D18 C2×Dic9 C3×C3⋊D4 Dic9 C2×C18 C3×Dic3 S3×C6 C62 C3×C9 C3⋊D4 Dic3 D6 C2×C6 C9 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 3 3 3 3 4 1 1 1 3 3 2 3 6

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

 27 0 0 0 33 11 0 0 0 0 1 0 0 0 0 1
,
 29 5 0 0 24 8 0 0 0 0 1 0 0 0 0 1
,
 29 5 0 0 17 8 0 0 0 0 12 0 0 0 0 34
,
 26 30 0 0 28 11 0 0 0 0 0 34 0 0 12 0
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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