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## G = Dic18⋊S3order 432 = 24·33

### 3rd semidirect product of Dic18 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Dic18⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C18.D6 — Dic18⋊S3
 Lower central C3×C9 — C3×C18 — Dic18⋊S3
 Upper central C1 — C2 — C4

Generators and relations for Dic18⋊S3
G = < a,b,c,d | a36=c3=d2=1, b2=a18, bab-1=a-1, ac=ca, dad=a17, bc=cb, bd=db, dcd=c-1 >

Subgroups: 784 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×Q8, D9, C18, C18, C3⋊S3, C3×C6, Dic6, Dic6, C4×S3, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, D18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C9⋊S3, C3×C18, Dic18, Dic18, C4×D9, Q8×C9, C6.D6, C322Q8, C3×Dic6, C3×Dic6, C4×C3⋊S3, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, C2×C9⋊S3, Q8×D9, Dic3.D6, C9⋊Dic6, C18.D6, C3×Dic18, C9×Dic6, C4×C9⋊S3, Dic18⋊S3
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, C22×S3, D18, S32, S3×Q8, C22×D9, C2×S32, S3×D9, Q8×D9, Dic3.D6, C2×S3×D9, Dic18⋊S3

Smallest permutation representation of Dic18⋊S3
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 55 19 37)(2 54 20 72)(3 53 21 71)(4 52 22 70)(5 51 23 69)(6 50 24 68)(7 49 25 67)(8 48 26 66)(9 47 27 65)(10 46 28 64)(11 45 29 63)(12 44 30 62)(13 43 31 61)(14 42 32 60)(15 41 33 59)(16 40 34 58)(17 39 35 57)(18 38 36 56)
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)
(1 19)(2 36)(3 17)(4 34)(5 15)(6 32)(7 13)(8 30)(9 11)(10 28)(12 26)(14 24)(16 22)(18 20)(21 35)(23 33)(25 31)(27 29)(37 55)(38 72)(39 53)(40 70)(41 51)(42 68)(43 49)(44 66)(45 47)(46 64)(48 62)(50 60)(52 58)(54 56)(57 71)(59 69)(61 67)(63 65)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A 18B 18C 18D 18E 18F 36A ··· 36I 36J ··· 36O order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 18 18 18 18 18 36 ··· 36 36 ··· 36 size 1 1 27 27 2 2 4 2 6 6 18 18 54 2 2 4 2 2 2 4 4 4 4 4 4 4 12 12 36 36 2 2 2 4 4 4 4 ··· 4 12 ··· 12

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + + + - - + + - + image C1 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D9 D18 D18 S32 S3×Q8 S3×Q8 C2×S32 S3×D9 Q8×D9 Dic3.D6 C2×S3×D9 Dic18⋊S3 kernel Dic18⋊S3 C9⋊Dic6 C18.D6 C3×Dic18 C9×Dic6 C4×C9⋊S3 Dic18 C3×Dic6 C9⋊S3 Dic9 C36 C3×Dic3 C3×C12 Dic6 Dic3 C12 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 2 1 2 1 3 6 3 1 1 1 1 3 3 2 3 6

Matrix representation of Dic18⋊S3 in GL6(𝔽37)

 0 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 20 0 0 0 0 17 26
,
 3 8 0 0 0 0 8 34 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,17,0,0,0,0,20,26],[3,8,0,0,0,0,8,34,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic18⋊S3 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes S_3
% in TeX

G:=Group("Dic18:S3");
// GroupNames label

G:=SmallGroup(432,283);
// by ID

G=gap.SmallGroup(432,283);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^36=c^3=d^2=1,b^2=a^18,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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