metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D18, D72⋊2C2, C24.1D6, C72⋊1C22, C4.14D36, C36.12D4, C12.7D12, D36⋊4C22, M4(2)⋊1D9, C22.5D36, C36.32C23, Dic18⋊4C22, (C2×D36)⋊7C2, C72⋊C2⋊1C2, C9⋊1(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), D36⋊5C2⋊2C2, (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
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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, D9, C18, C18, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C22×S3, C8⋊C22, Dic9, C36, D18, C2×C18, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C72, Dic18, C4×D9, D36, D36, D36, C9⋊D4, C2×C36, C22×D9, C8⋊D6, C72⋊C2, D72, C9×M4(2), C2×D36, D36⋊5C2, C8⋊D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, C8⋊C22, D18, C2×D12, D36, C22×D9, C8⋊D6, C2×D36, C8⋊D18
►On 72 points
(1 37 30 65 16 46 20 56)(2 47 31 57 17 38 21 66)(3 39 32 67 18 48 22 58)(4 49 33 59 10 40 23 68)(5 41 34 69 11 50 24 60)(6 51 35 61 12 42 25 70)(7 43 36 71 13 52 26 62)(8 53 28 63 14 44 27 72)(9 45 29 55 15 54 19 64)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(55 64)(56 63)(57 62)(58 61)(59 60)(65 72)(66 71)(67 70)(68 69)
G:=sub<Sym(72)| (1,37,30,65,16,46,20,56)(2,47,31,57,17,38,21,66)(3,39,32,67,18,48,22,58)(4,49,33,59,10,40,23,68)(5,41,34,69,11,50,24,60)(6,51,35,61,12,42,25,70)(7,43,36,71,13,52,26,62)(8,53,28,63,14,44,27,72)(9,45,29,55,15,54,19,64), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(55,64)(56,63)(57,62)(58,61)(59,60)(65,72)(66,71)(67,70)(68,69)>;
G:=Group( (1,37,30,65,16,46,20,56)(2,47,31,57,17,38,21,66)(3,39,32,67,18,48,22,58)(4,49,33,59,10,40,23,68)(5,41,34,69,11,50,24,60)(6,51,35,61,12,42,25,70)(7,43,36,71,13,52,26,62)(8,53,28,63,14,44,27,72)(9,45,29,55,15,54,19,64), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(55,64)(56,63)(57,62)(58,61)(59,60)(65,72)(66,71)(67,70)(68,69) );
G=PermutationGroup([[(1,37,30,65,16,46,20,56),(2,47,31,57,17,38,21,66),(3,39,32,67,18,48,22,58),(4,49,33,59,10,40,23,68),(5,41,34,69,11,50,24,60),(6,51,35,61,12,42,25,70),(7,43,36,71,13,52,26,62),(8,53,28,63,14,44,27,72),(9,45,29,55,15,54,19,64)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,36),(18,35),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(55,64),(56,63),(57,62),(58,61),(59,60),(65,72),(66,71),(67,70),(68,69)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | 36H | 36I | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 36 | 36 | 36 | 2 | 2 | 2 | 36 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | D12 | D12 | D18 | D18 | D36 | D36 | C8⋊C22 | C8⋊D6 | C8⋊D18 |
| kernel | C8⋊D18 | C72⋊C2 | D72 | C9×M4(2) | C2×D36 | D36⋊5C2 | C3×M4(2) | C36 | C2×C18 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C8⋊D18 ►in GL4(𝔽73) generated by
| 23 | 41 | 26 | 15 |
| 45 | 12 | 58 | 41 |
| 31 | 9 | 35 | 13 |
| 26 | 60 | 67 | 3 |
| 42 | 3 | 0 | 0 |
| 70 | 45 | 0 | 0 |
| 57 | 38 | 31 | 70 |
| 49 | 30 | 3 | 28 |
| 42 | 28 | 0 | 0 |
| 70 | 31 | 0 | 0 |
| 2 | 45 | 59 | 7 |
| 53 | 60 | 66 | 14 |
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