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G = D18⋊C12  order 432 = 24·33

The semidirect product of D18 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: D18⋊C12, C62.25D6, D18⋊C4⋊C3, (C2×C36)⋊6C6, (C6×C12).3S3, C18.5(C3×D4), (C22×D9).C6, C6.19(S3×C12), C18.4(C2×C12), (C2×Dic9)⋊1C6, (C3×C6).22D12, C6.15(C3×D12), C32.(D6⋊C4), C2.2(D36⋊C3), C2.2(Dic9⋊C6), (C2×3- 1+2).5D4, 3- 1+2⋊1(C22⋊C4), (C22×3- 1+2).5C22, (C2×C9⋊C6)⋊C4, C2.5(C4×C9⋊C6), (C2×C9⋊C12)⋊1C2, (C2×C4)⋊1(C9⋊C6), C9⋊1(C3×C22⋊C4), C3.3(C3×D6⋊C4), (C22×C9⋊C6).C2, (C2×C12).8(C3×S3), (C2×C18).5(C2×C6), (C2×C6).45(S3×C6), (C3×C6).19(C4×S3), C6.17(C3×C3⋊D4), C22.6(C2×C9⋊C6), (C3×C6).22(C3⋊D4), (C2×C4×3- 1+2)⋊6C2, (C2×3- 1+2).4(C2×C4), SmallGroup(432,147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — D18⋊C12
Chief series
C1 — C3 — C9 — C18 — C2×C18 — C22×3- 1+2 — C22×C9⋊C6 — D18⋊C12
Lower central
C9 — C18 — D18⋊C12
Upper central
C1 — C22 — C2×C4

Generators and relations for D18⋊C12
 G = < a,b,c | a18=b2=c12=1, bab=a-1, cac-1=a13, cbc-1=a3b >

Subgroups: 470 in 110 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, 3- 1+2, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, D6⋊C4, C3×C22⋊C4, C9⋊C6, C2×3- 1+2, C2×Dic9, C2×C36, C2×C36, C22×D9, C6×Dic3, C6×C12, S3×C2×C6, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, D18⋊C4, C3×D6⋊C4, C2×C9⋊C12, C2×C4×3- 1+2, C22×C9⋊C6, D18⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, S3×C6, D6⋊C4, C3×C22⋊C4, C9⋊C6, S3×C12, C3×D12, C3×C3⋊D4, C2×C9⋊C6, C3×D6⋊C4, C4×C9⋊C6, D36⋊C3, Dic9⋊C6, D18⋊C12

Smallest permutation representation of D18⋊C12
►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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 36)(34 35)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(52 54)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)

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

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D···6I6J6K6L6M9A9B9C12A12B12C12D12E12F12G12H12I12J12K12L18A···18I36A···36L
order12222233344446666···6666699912121212121212121212121218···1836···36
size111118182332218182223···31818181866622226666181818186···66···6

62 irreducible representations

dim111111111122222222222266666
type+++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D4D6C3×S3C3×D4C4×S3D12C3⋊D4S3×C6S3×C12C3×D12C3×C3⋊D4C9⋊C6C2×C9⋊C6C4×C9⋊C6D36⋊C3Dic9⋊C6
kernelD18⋊C12C2×C9⋊C12C2×C4×3- 1+2C22×C9⋊C6D18⋊C4C2×C9⋊C6C2×Dic9C2×C36C22×D9D18C6×C12C2×3- 1+2C62C2×C12C18C3×C6C3×C6C3×C6C2×C6C6C6C6C2×C4C22C2C2C2
# reps111124222812124222244411222

Matrix representation of D18⋊C12 ►in GL8(𝔽37)

036000000
11000000
00000100
0000363600
00000001
0000003636
00100000
00010000
,
11000000
036000000
0000363600
00000100
0036360000
00010000
00000010
0000003636
,
3431000000
63000000
003100000
000310000
00006600
000031000
000000031
00000066

G:=sub<GL(8,GF(37))| [0,1,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0],[1,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,36],[34,6,0,0,0,0,0,0,31,3,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,6,31,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,31,6] >;
 

D18⋊C12 in GAP, Magma, Sage, TeX 

D_{18}\rtimes C_{12}
 
% in TeX
 
G:=Group("D18:C12");
 
// GroupNames label
 
G:=SmallGroup(432,147);
 
// by ID
 
G=gap.SmallGroup(432,147);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,10085,2035,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c|a^18=b^2=c^12=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^3*b>;
 
// generators/relations
 

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