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## G = Dic18⋊C6order 432 = 24·33

### 1st semidirect product of Dic18 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — Dic18⋊C6
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C36.C6 — Dic18⋊C6
 Lower central C9 — C18 — C36 — Dic18⋊C6
 Upper central C1 — C2 — C4 — D4

Generators and relations for Dic18⋊C6
G = < a,b,c | a36=c6=1, b2=a18, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Subgroups: 230 in 68 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, SD16, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×Q8, 3- 1+2, Dic9, C36, C36, C2×C18, C3×Dic3, C3×C12, C62, D4.S3, C3×SD16, C2×3- 1+2, C2×3- 1+2, C9⋊C8, Dic18, D4×C9, D4×C9, C3×C3⋊C8, C3×Dic6, D4×C32, C9⋊C12, C4×3- 1+2, C22×3- 1+2, D4.D9, C3×D4.S3, C9⋊C24, C36.C6, D4×3- 1+2, Dic18⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊D4, C3×D4, S3×C6, D4.S3, C3×SD16, C9⋊C6, C3×C3⋊D4, C2×C9⋊C6, C3×D4.S3, Dic9⋊C6, Dic18⋊C6

Smallest permutation representation of Dic18⋊C6
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 64 19 46)(2 63 20 45)(3 62 21 44)(4 61 22 43)(5 60 23 42)(6 59 24 41)(7 58 25 40)(8 57 26 39)(9 56 27 38)(10 55 28 37)(11 54 29 72)(12 53 30 71)(13 52 31 70)(14 51 32 69)(15 50 33 68)(16 49 34 67)(17 48 35 66)(18 47 36 65)
(2 32 26 20 14 8)(3 27 15)(4 22)(5 17 29)(6 12 18 24 30 36)(9 33 21)(10 28)(11 23 35)(16 34)(37 46)(38 41 62 65 50 53)(39 72 51 48 63 60)(40 67)(42 57 54 69 66 45)(43 52)(44 47 68 71 56 59)(49 58)(55 64)(61 70)

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

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

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

41 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18I 24A 24B 24C 24D 36A 36B 36C order 1 2 2 3 3 3 4 4 6 6 6 6 6 6 6 8 8 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 24 24 24 24 36 36 36 size 1 1 4 2 3 3 2 36 2 3 3 4 4 12 12 18 18 6 6 6 4 6 6 36 36 6 6 6 12 ··· 12 18 18 18 18 12 12 12

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + - + + + - + + image C1 C2 C2 C2 C3 C6 C6 C6 Dic18⋊C6 S3 D4 D6 SD16 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×SD16 C3×C3⋊D4 D4.S3 C3×D4.S3 C9⋊C6 C2×C9⋊C6 Dic9⋊C6 kernel Dic18⋊C6 C9⋊C24 C36.C6 D4×3- 1+2 D4.D9 C9⋊C8 Dic18 D4×C9 C1 D4×C32 C2×3- 1+2 C3×C12 3- 1+2 C3×D4 C18 C3×C6 C12 C9 C6 C32 C3 D4 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of Dic18⋊C6 in GL10(𝔽73)

 72 1 70 3 0 0 0 0 0 0 72 0 70 0 0 0 0 0 0 0 25 48 1 72 0 0 0 0 0 0 25 0 1 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 0 11 57 0 0 0 9 0 0 0 0 2 28 62 65 0 0 0 0 0 0 0 62 47 0 65 0
,
 0 0 31 34 0 0 0 0 0 0 0 0 65 42 0 0 0 0 0 0 15 40 0 0 0 0 0 0 0 0 55 58 0 0 0 0 0 0 0 0 0 0 0 0 10 66 21 59 0 0 0 0 0 0 66 21 43 0 59 0 0 0 0 0 21 43 66 0 0 59 0 0 0 0 16 49 56 63 7 52 0 0 0 0 49 21 69 7 52 30 0 0 0 0 56 69 21 52 30 7
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 48 0 72 0 0 0 0 0 0 0 0 48 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 64 32 61 72 0 0 0 0 0 0 32 24 23 0 65 0 0 0 0 0 61 23 9 0 0 9

G:=sub<GL(10,GF(73))| [72,72,25,25,0,0,0,0,0,0,1,0,48,0,0,0,0,0,0,0,70,70,1,1,0,0,0,0,0,0,3,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,65,11,2,0,0,0,0,0,9,0,0,57,28,62,0,0,0,0,0,9,0,0,62,47,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,9,0,0],[0,0,15,55,0,0,0,0,0,0,0,0,40,58,0,0,0,0,0,0,31,65,0,0,0,0,0,0,0,0,34,42,0,0,0,0,0,0,0,0,0,0,0,0,10,66,21,16,49,56,0,0,0,0,66,21,43,49,21,69,0,0,0,0,21,43,66,56,69,21,0,0,0,0,59,0,0,63,7,52,0,0,0,0,0,59,0,7,52,30,0,0,0,0,0,0,59,52,30,7],[1,0,48,0,0,0,0,0,0,0,0,1,0,48,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,64,32,61,0,0,0,0,0,8,0,32,24,23,0,0,0,0,0,0,64,61,23,9,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,0,0,9] >;

Dic18⋊C6 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes C_6
% in TeX

G:=Group("Dic18:C6");
// GroupNames label

G:=SmallGroup(432,154);
// by ID

G=gap.SmallGroup(432,154);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^36=c^6=1,b^2=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations

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