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G = Dic18⋊C6  order 432 = 24·33

1st semidirect product of Dic18 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: Dic18⋊1C6, 3- 1+2⋊2SD16, C9⋊C8⋊1C6, D4.D9⋊C3, C9⋊C24⋊1C2, D4.(C9⋊C6), C12.5(S3×C6), C36.1(C2×C6), (D4×C9).1C6, C9⋊2(C3×SD16), C18.7(C3×D4), (C3×C12).10D6, C36.C6⋊1C2, (D4×C32).2S3, C32.(D4.S3), C2.4(Dic9⋊C6), (D4×3- 1+2).1C2, (C2×3- 1+2).7D4, (C4×3- 1+2).1C22, C4.1(C2×C9⋊C6), (C3×D4).5(C3×S3), C6.22(C3×C3⋊D4), C3.3(C3×D4.S3), (C3×C6).25(C3⋊D4), SmallGroup(432,154)

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 2A2B3A3B3C4A4B6A6B6C6D6E6F6G8A8B9A9B9C12A12B12C12D12E18A18B18C18D···18I24A24B24C24D36A36B36C
order12233344666666688999121212121218181818···1824242424363636
size1142332362334412121818666466363666612···1218181818121212

41 irreducible representations

dim1111111112222222222244666
type++++-+++-++
imageC1C2C2C2C3C6C6C6Dic18⋊C6S3D4D6SD16C3×S3C3×D4C3⋊D4S3×C6C3×SD16C3×C3⋊D4D4.S3C3×D4.S3C9⋊C6C2×C9⋊C6Dic9⋊C6
kernelDic18⋊C6C9⋊C24C36.C6D4×3- 1+2D4.D9C9⋊C8Dic18D4×C9C1D4×C32C2×3- 1+2C3×C123- 1+2C3×D4C18C3×C6C12C9C6C32C3D4C4C2
# reps111122221111222224412112

Matrix representation of Dic18⋊C6 ►in GL10(𝔽73)

721703000000
720700000000
2548172000000
25010000000
0000090000
0000009000
00006500000
000011570009
0000228626500
0000062470650
,
003134000000
006542000000
154000000000
555800000000
00001066215900
00006621430590
00002143660059
000016495663752
000049216975230
000056692152307
,
1000000000
0100000000
480720000000
048072000000
0000100000
0000080000
00000064000
00006432617200
00003224230650
000061239009

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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