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G = D18⋊C4order 144 = 24·32

The semidirect product of D18 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D18⋊C4, C2.2D36, C18.6D4, C6.5D12, C22.6D18, (C2×C4)⋊1D9, (C2×C36)⋊1C2, C6.9(C4×S3), C3.(D6⋊C4), C2.5(C4×D9), C91(C22⋊C4), (C2×C12).2S3, C18.5(C2×C4), (C2×C6).22D6, (C22×D9).C2, (C2×Dic9)⋊1C2, C2.2(C9⋊D4), C6.13(C3⋊D4), (C2×C18).6C22, SmallGroup(144,14)

Series: Derived Chief Lower central Upper central

C1C18 — D18⋊C4
C1C3C9C18C2×C18C22×D9 — D18⋊C4
C9C18 — D18⋊C4
C1C22C2×C4

Generators and relations for D18⋊C4
 G = < a,b,c | a18=b2=c4=1, bab=a-1, ac=ca, cbc-1=a9b >

18C2
18C2
2C4
9C22
9C22
18C4
18C22
18C22
6S3
6S3
9C23
9C2×C4
2C12
3D6
3D6
6Dic3
6D6
6D6
2D9
2D9
9C22⋊C4
3C2×Dic3
3C22×S3
2C36
2D18
2D18
2Dic9
3D6⋊C4

Smallest permutation representation of D18⋊C4
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 20)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(55 65)(56 64)(57 63)(58 62)(59 61)(66 72)(67 71)(68 70)
(1 65 29 51)(2 66 30 52)(3 67 31 53)(4 68 32 54)(5 69 33 37)(6 70 34 38)(7 71 35 39)(8 72 36 40)(9 55 19 41)(10 56 20 42)(11 57 21 43)(12 58 22 44)(13 59 23 45)(14 60 24 46)(15 61 25 47)(16 62 26 48)(17 63 27 49)(18 64 28 50)

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

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

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

D18⋊C4 is a maximal subgroup of
C422D9  C4×D36  C427D9  C423D9  C22⋊C4×D9  Dic94D4  C223D36  C23.9D18  D18⋊D4  Dic9.D4  C22.4D36  C4⋊C47D9  D36⋊C4  D18.D4  C4⋊D36  D18⋊Q8  D182Q8  C4⋊C4⋊D9  C4×C9⋊D4  C23.28D18  C367D4  C232D18  Dic9⋊D4  D183Q8  C36.23D4  D54⋊C4  D18⋊Dic3  C6.18D36  D18⋊C12  C6.11D36
D18⋊C4 is a maximal quotient of
C424D9  C22.D36  C18.Q16  C18.D8  C36.45D4  D18⋊C8  C2.D72  C4.D36  C36.48D4  Dic18⋊C4  C18.C42  D54⋊C4  D18⋊Dic3  C6.18D36  C6.11D36

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122222344446669991212121218···1836···36
size11111818222181822222222222···22···2

42 irreducible representations

dim1111122222222222
type+++++++++++
imageC1C2C2C2C4S3D4D6D9C4×S3D12C3⋊D4D18C4×D9D36C9⋊D4
kernelD18⋊C4C2×Dic9C2×C36C22×D9D18C2×C12C18C2×C6C2×C4C6C6C6C22C2C2C2
# reps1111412132223666

Matrix representation of D18⋊C4 in GL3(𝔽37) generated by

100
0617
02026
,
3600
0617
01131
,
600
0510
02732
G:=sub<GL(3,GF(37))| [1,0,0,0,6,20,0,17,26],[36,0,0,0,6,11,0,17,31],[6,0,0,0,5,27,0,10,32] >;

D18⋊C4 in GAP, Magma, Sage, TeX

D_{18}\rtimes C_4
% in TeX

G:=Group("D18:C4");
// GroupNames label

G:=SmallGroup(144,14);
// by ID

G=gap.SmallGroup(144,14);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^18=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations

Export

Subgroup lattice of D18⋊C4 in TeX

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