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G = Dic18⋊2C6  order 432 = 24·33

2nd semidirect product of Dic18 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: Dic18⋊2C6, C62.15D6, D4⋊2D9⋊C3, (D4×C9)⋊3C6, (C4×D9)⋊2C6, C9⋊D4⋊2C6, D4⋊2(C9⋊C6), C36.5(C2×C6), C12.18(S3×C6), Dic9⋊C6⋊2C2, (C2×Dic9)⋊3C6, D18.2(C2×C6), (C3×C12).27D6, C36.C6⋊2C2, C9⋊C12.3C22, C18.6(C22×C6), Dic9.3(C2×C6), (D4×C32).6S3, C32.(D4⋊2S3), (D4×3- 1+2)⋊3C2, 3- 1+2⋊2(C4○D4), (C4×3- 1+2).5C22, (C2×3- 1+2).6C23, (C22×3- 1+2).C22, (C4×C9⋊C6)⋊2C2, C4.5(C2×C9⋊C6), C9⋊2(C3×C4○D4), (C2×C18).(C2×C6), C6.36(S3×C2×C6), (C2×C9⋊C12)⋊3C2, (C2×C6).13(S3×C6), C2.7(C22×C9⋊C6), C22.1(C2×C9⋊C6), (C3×D4).12(C3×S3), C3.3(C3×D4⋊2S3), (C2×C9⋊C6).2C22, (C3×C6).30(C22×S3), SmallGroup(432,363)

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 Dic18⋊2C6
 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, D4⋊2S3, 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, D4⋊2D9, C3×D4⋊2S3, C36.C6, C4×C9⋊C6, C2×C9⋊C12, Dic9⋊C6, D4×3- 1+2, Dic18⋊2C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, D4⋊2S3, C3×C4○D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×D4⋊2S3, C22×C9⋊C6, Dic18⋊2C6

Smallest permutation representation of Dic18⋊2C6
►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 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K9A9B9C12A12B12C12D12E12F12G12H12I12J12K18A18B18C18D···18I36A36B36C
order122223334444466666666666999121212121212121212121218181818···18363636
size1122182332991818233446666181866646699991818181866612···12121212

50 irreducible representations

dim111111111111122222222244666
type++++++-+++-+++
imageC1C2C2C2C2C2C3C6C6C6C6C6Dic18⋊2C6S3D6D6C4○D4C3×S3S3×C6S3×C6C3×C4○D4D4⋊2S3C3×D4⋊2S3C9⋊C6C2×C9⋊C6C2×C9⋊C6
kernelDic18⋊2C6C36.C6C4×C9⋊C6C2×C9⋊C12Dic9⋊C6D4×3- 1+2D4⋊2D9Dic18C4×D9C2×Dic9C9⋊D4D4×C9C1D4×C32C3×C12C623- 1+2C3×D4C12C2×C6C9C32C3D4C4C22
# reps11122122244211122224412112

Matrix representation of Dic18⋊2C6 ►in GL10(𝔽37)

0001000000
003636000000
03600000000
1100000000
00000003600
00000013600
00000000036
00000000136
00003610000
00003600000
,
00310000000
0066000000
31000000000
6600000000
000033290000
00002540000
000000001233
00000000825
000000123300
00000082500
,
26000000000
02600000000
00110000000
00011000000
0000100000
0000010000
00000036100
00000036000
00000000036
00000000136

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] >;
 

Dic18⋊2C6 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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