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G = D46D18order 288 = 25·32

2nd semidirect product of D4 and D18 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46D18, C233D18, D368C22, C18.7C24, C912+ 1+4, C36.21C23, D18.3C23, Dic188C22, Dic9.4C23, (D4×D9)⋊4C2, (C2×D4)⋊7D9, (C2×C4)⋊3D18, (D4×C18)⋊7C2, D42D94C2, C3.(D46D6), (C2×C36)⋊3C22, (C6×D4).14S3, (C3×D4).37D6, (C4×D9)⋊1C22, (D4×C9)⋊7C22, C9⋊D43C22, D365C25C2, C2.8(C23×D9), (C2×C12).100D6, (C2×C18).2C23, C6.44(S3×C23), C4.21(C22×D9), (C22×C6).60D6, C12.61(C22×S3), (C22×C18)⋊5C22, (C2×Dic9)⋊4C22, (C22×D9)⋊3C22, C22.6(C22×D9), (C2×C9⋊D4)⋊11C2, (C2×C6).223(C22×S3), SmallGroup(288,358)

Series: Derived Chief Lower central Upper central

C1C18 — D46D18
C1C3C9C18D18C22×D9D4×D9 — D46D18
C9C18 — D46D18
C1C2C2×D4

Generators and relations for D46D18
 G = < a,b,c,d | a4=b2=c18=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1080 in 249 conjugacy classes, 102 normal (16 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C9, Dic3 [×4], C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C2×C6 [×2], C2×D4, C2×D4 [×8], C4○D4 [×6], D9 [×4], C18, C18 [×5], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], 2+ 1+4, Dic9 [×4], C36 [×2], D18 [×4], D18 [×4], C2×C18, C2×C18 [×4], C2×C18 [×2], C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, Dic18 [×2], C4×D9 [×4], D36 [×2], C2×Dic9 [×4], C9⋊D4 [×12], C2×C36, D4×C9 [×4], C22×D9 [×4], C22×C18 [×2], D46D6, D365C2 [×2], D4×D9 [×4], D42D9 [×4], C2×C9⋊D4 [×4], D4×C18, D46D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2+ 1+4, D18 [×7], S3×C23, C22×D9 [×7], D46D6, C23×D9, D46D18

Smallest permutation representation of D46D18
On 72 points
Generators in S72
(1 72 10 63)(2 64 11 55)(3 56 12 65)(4 66 13 57)(5 58 14 67)(6 68 15 59)(7 60 16 69)(8 70 17 61)(9 62 18 71)(19 54 28 45)(20 46 29 37)(21 38 30 47)(22 48 31 39)(23 40 32 49)(24 50 33 41)(25 42 34 51)(26 52 35 43)(27 44 36 53)
(1 45)(2 37)(3 47)(4 39)(5 49)(6 41)(7 51)(8 43)(9 53)(10 54)(11 46)(12 38)(13 48)(14 40)(15 50)(16 42)(17 52)(18 44)(19 63)(20 55)(21 65)(22 57)(23 67)(24 59)(25 69)(26 61)(27 71)(28 72)(29 64)(30 56)(31 66)(32 58)(33 68)(34 60)(35 70)(36 62)
(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 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(53 54)(55 61)(56 60)(57 59)(62 72)(63 71)(64 70)(65 69)(66 68)

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

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

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

57 conjugacy classes

class 1 2A2B···2F2G2H2I2J 3 4A4B4C4D4E4F6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order122···2222234444446666666999121218···1818···1836···36
size112···218181818222181818182224444222442···24···44···4

57 irreducible representations

dim11111122222222444
type+++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6D9D18D18D182+ 1+4D46D6D46D18
kernelD46D18D365C2D4×D9D42D9C2×C9⋊D4D4×C18C6×D4C2×C12C3×D4C22×C6C2×D4C2×C4D4C23C9C3C1
# reps124441114233126126

Matrix representation of D46D18 in GL6(𝔽37)

100000
010000
00003023
0000147
0071400
00233000
,
100000
010000
000010
000001
001000
000100
,
6170000
20260000
000100
00363600
0000036
000011
,
6170000
11310000
000100
001000
0000036
0000360

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,23,0,0,0,0,14,30,0,0,30,14,0,0,0,0,23,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,20,0,0,0,0,17,26,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1],[6,11,0,0,0,0,17,31,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0] >;

D46D18 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{18}
% in TeX

G:=Group("D4:6D18");
// GroupNames label

G:=SmallGroup(288,358);
// by ID

G=gap.SmallGroup(288,358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,6725,292,9414]);
// Polycyclic

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

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