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G = C8⋊D18order 288 = 25·32

1st semidirect product of C8 and D18 acting via D18/C9=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C81D18, D722C2, C24.1D6, C721C22, C4.14D36, C36.12D4, C12.7D12, D364C22, M4(2)⋊1D9, C22.5D36, C36.32C23, Dic184C22, (C2×D36)⋊7C2, C72⋊C21C2, C91(C8⋊C22), C3.(C8⋊D6), (C2×C18).5D4, (C2×C6).6D12, C18.13(C2×D4), (C2×C4).12D18, (C2×C12).53D6, C6.42(C2×D12), C2.15(C2×D36), D365C22C2, (C9×M4(2))⋊1C2, C4.30(C22×D9), (C2×C36).31C22, (C3×M4(2)).1S3, C12.183(C22×S3), SmallGroup(288,118)

Series: Derived Chief Lower central Upper central

C1C36 — C8⋊D18
C1C3C9C18C36D36C2×D36 — C8⋊D18
C9C18C36 — C8⋊D18
C1C2C2×C4M4(2)

Generators and relations for C8⋊D18
 G = < a,b,c | a8=b18=c2=1, bab-1=a5, cac=a3, cbc=b-1 >

Subgroups: 644 in 102 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C9, Dic3, C12 [×2], D6 [×5], C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D9 [×3], C18, C18, C24 [×2], Dic6, C4×S3, D12 [×4], C3⋊D4, C2×C12, C22×S3, C8⋊C22, Dic9, C36 [×2], D18 [×5], C2×C18, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, C72 [×2], Dic18, C4×D9, D36, D36 [×2], D36, C9⋊D4, C2×C36, C22×D9, C8⋊D6, C72⋊C2 [×2], D72 [×2], C9×M4(2), C2×D36, D365C2, C8⋊D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, D12 [×2], C22×S3, C8⋊C22, D18 [×3], C2×D12, D36 [×2], C22×D9, C8⋊D6, C2×D36, C8⋊D18

Smallest permutation representation of C8⋊D18
On 72 points
Generators in S72
(1 39 18 55 31 48 22 64)(2 49 10 65 32 40 23 56)(3 41 11 57 33 50 24 66)(4 51 12 67 34 42 25 58)(5 43 13 59 35 52 26 68)(6 53 14 69 36 44 27 60)(7 45 15 61 28 54 19 70)(8 37 16 71 29 46 20 62)(9 47 17 63 30 38 21 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 19)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(37 47)(38 46)(39 45)(40 44)(41 43)(48 54)(49 53)(50 52)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(71 72)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B8A8B9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122222344466889991212121818181818182424242436···3636363672···72
size11236363622236244422222422244444442···24444···4

51 irreducible representations

dim111111222222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D9D12D12D18D18D36D36C8⋊C22C8⋊D6C8⋊D18
kernelC8⋊D18C72⋊C2D72C9×M4(2)C2×D36D365C2C3×M4(2)C36C2×C18C24C2×C12M4(2)C12C2×C6C8C2×C4C4C22C9C3C1
# reps122111111213226366126

Matrix representation of C8⋊D18 in GL4(𝔽73) generated by

23412615
45125841
3193513
2660673
,
42300
704500
57383170
4930328
,
422800
703100
245597
53606614
G:=sub<GL(4,GF(73))| [23,45,31,26,41,12,9,60,26,58,35,67,15,41,13,3],[42,70,57,49,3,45,38,30,0,0,31,3,0,0,70,28],[42,70,2,53,28,31,45,60,0,0,59,66,0,0,7,14] >;

C8⋊D18 in GAP, Magma, Sage, TeX

C_8\rtimes D_{18}
% in TeX

G:=Group("C8:D18");
// GroupNames label

G:=SmallGroup(288,118);
// by ID

G=gap.SmallGroup(288,118);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^8=b^18=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

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