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## G = Dic18⋊2C6order 432 = 24·33

### 2nd semidirect product of Dic18 and C6 acting faithfully

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic182C6
G = < a,b,c | a36=c6=1, b2=a18, bab-1=a-1, cac-1=a7, cbc-1=a18b >

Subgroups: 438 in 128 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, 3- 1+2, Dic9, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, D42S3, C3×C4○D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, D4×C9, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, C9⋊C12, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C22×3- 1+2, D42D9, C3×D42S3, C36.C6, C4×C9⋊C6, C2×C9⋊C12, Dic9⋊C6, D4×3- 1+2, Dic182C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, D42S3, C3×C4○D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×D42S3, C22×C9⋊C6, Dic182C6

Smallest permutation representation of Dic182C6
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 37 19 55)(2 72 20 54)(3 71 21 53)(4 70 22 52)(5 69 23 51)(6 68 24 50)(7 67 25 49)(8 66 26 48)(9 65 27 47)(10 64 28 46)(11 63 29 45)(12 62 30 44)(13 61 31 43)(14 60 32 42)(15 59 33 41)(16 58 34 40)(17 57 35 39)(18 56 36 38)
(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 55)(38 50 62)(39 45 51 57 63 69)(41 71 65 59 53 47)(42 66 54)(43 61)(44 56 68)(48 72 60)(49 67)

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

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

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

50 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 6J 6K 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 18A 18B 18C 18D ··· 18I 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 6 6 9 9 9 12 12 12 12 12 12 12 12 12 12 12 18 18 18 18 ··· 18 36 36 36 size 1 1 2 2 18 2 3 3 2 9 9 18 18 2 3 3 4 4 6 6 6 6 18 18 6 6 6 4 6 6 9 9 9 9 18 18 18 18 6 6 6 12 ··· 12 12 12 12

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + + + - + + + - + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 Dic18⋊2C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 D4⋊2S3 C3×D4⋊2S3 C9⋊C6 C2×C9⋊C6 C2×C9⋊C6 kernel Dic18⋊2C6 C36.C6 C4×C9⋊C6 C2×C9⋊C12 Dic9⋊C6 D4×3- 1+2 D4⋊2D9 Dic18 C4×D9 C2×Dic9 C9⋊D4 D4×C9 C1 D4×C32 C3×C12 C62 3- 1+2 C3×D4 C12 C2×C6 C9 C32 C3 D4 C4 C22 # reps 1 1 1 2 2 1 2 2 2 4 4 2 1 1 1 2 2 2 2 4 4 1 2 1 1 2

Matrix representation of Dic182C6 in GL10(𝔽37)

 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 36 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0
,
 0 0 31 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 33 29 0 0 0 0 0 0 0 0 25 4 0 0 0 0 0 0 0 0 0 0 0 0 12 33 0 0 0 0 0 0 0 0 8 25 0 0 0 0 0 0 12 33 0 0 0 0 0 0 0 0 8 25 0 0
,
 26 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 1 36

G:=sub<GL(10,GF(37))| [0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,36,0,0],[0,0,31,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,31,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,33,25,0,0,0,0,0,0,0,0,29,4,0,0,0,0,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,0,33,25,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,0,33,25,0,0],[26,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,36] >;

Dic182C6 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes_2C_6
% in TeX

G:=Group("Dic18:2C6");
// GroupNames label

G:=SmallGroup(432,363);
// by ID

G=gap.SmallGroup(432,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,10085,1034,292,14118]);
// 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^18*b>;
// generators/relations

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