metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36⋊4C4, C36.54D4, Dic18⋊4C4, M4(2)⋊4D9, C22.3D36, C9⋊2C4≀C2, C4.3(C4×D9), C12.6(C4×S3), C36.6(C2×C4), (C2×C18).1D4, (C2×C6).3D12, (C4×Dic9)⋊1C2, (C2×C4).40D18, (C2×C12).44D6, C3.(D12⋊C4), C6.17(D6⋊C4), C4.29(C9⋊D4), (C9×M4(2))⋊8C2, C2.11(D18⋊C4), D36⋊5C2.2C2, (C2×C36).22C22, (C3×M4(2)).8S3, C12.124(C3⋊D4), C18.10(C22⋊C4), SmallGroup(288,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic18⋊C4
G = < a,b,c | a36=c4=1, b2=a18, bab-1=a-1, cac-1=a17, cbc-1=a27b >
Subgroups: 332 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C42, M4(2), C4○D4, D9, C18, C18, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4≀C2, Dic9, C36, D18, C2×C18, C4×Dic3, C3×M4(2), C4○D12, C72, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D12⋊C4, C4×Dic9, C9×M4(2), D36⋊5C2, Dic18⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4≀C2, D18, D6⋊C4, C4×D9, D36, C9⋊D4, D12⋊C4, D18⋊C4, Dic18⋊C4
►On 72 points
(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 37 19 55)(2 72 20 54)(3 71 21 53)(4 70 22 52)(5 69 23 51)(6 68 24 50)(7 67 25 49)(8 66 26 48)(9 65 27 47)(10 64 28 46)(11 63 29 45)(12 62 30 44)(13 61 31 43)(14 60 32 42)(15 59 33 41)(16 58 34 40)(17 57 35 39)(18 56 36 38)
(1 28 19 10)(2 9 20 27)(3 26 21 8)(4 7 22 25)(5 24 23 6)(11 18 29 36)(12 35 30 17)(13 16 31 34)(14 33 32 15)(38 54)(39 71)(40 52)(41 69)(42 50)(43 67)(44 48)(45 65)(47 63)(49 61)(51 59)(53 57)(56 72)(58 70)(60 68)(62 66)
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), (1,37,19,55)(2,72,20,54)(3,71,21,53)(4,70,22,52)(5,69,23,51)(6,68,24,50)(7,67,25,49)(8,66,26,48)(9,65,27,47)(10,64,28,46)(11,63,29,45)(12,62,30,44)(13,61,31,43)(14,60,32,42)(15,59,33,41)(16,58,34,40)(17,57,35,39)(18,56,36,38), (1,28,19,10)(2,9,20,27)(3,26,21,8)(4,7,22,25)(5,24,23,6)(11,18,29,36)(12,35,30,17)(13,16,31,34)(14,33,32,15)(38,54)(39,71)(40,52)(41,69)(42,50)(43,67)(44,48)(45,65)(47,63)(49,61)(51,59)(53,57)(56,72)(58,70)(60,68)(62,66)>;
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), (1,37,19,55)(2,72,20,54)(3,71,21,53)(4,70,22,52)(5,69,23,51)(6,68,24,50)(7,67,25,49)(8,66,26,48)(9,65,27,47)(10,64,28,46)(11,63,29,45)(12,62,30,44)(13,61,31,43)(14,60,32,42)(15,59,33,41)(16,58,34,40)(17,57,35,39)(18,56,36,38), (1,28,19,10)(2,9,20,27)(3,26,21,8)(4,7,22,25)(5,24,23,6)(11,18,29,36)(12,35,30,17)(13,16,31,34)(14,33,32,15)(38,54)(39,71)(40,52)(41,69)(42,50)(43,67)(44,48)(45,65)(47,63)(49,61)(51,59)(53,57)(56,72)(58,70)(60,68)(62,66) );
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)], [(1,37,19,55),(2,72,20,54),(3,71,21,53),(4,70,22,52),(5,69,23,51),(6,68,24,50),(7,67,25,49),(8,66,26,48),(9,65,27,47),(10,64,28,46),(11,63,29,45),(12,62,30,44),(13,61,31,43),(14,60,32,42),(15,59,33,41),(16,58,34,40),(17,57,35,39),(18,56,36,38)], [(1,28,19,10),(2,9,20,27),(3,26,21,8),(4,7,22,25),(5,24,23,6),(11,18,29,36),(12,35,30,17),(13,16,31,34),(14,33,32,15),(38,54),(39,71),(40,52),(41,69),(42,50),(43,67),(44,48),(45,65),(47,63),(49,61),(51,59),(53,57),(56,72),(58,70),(60,68),(62,66)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 3 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 1 | 1 | 2 | 18 | 18 | 18 | 18 | 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 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D9 | C4×S3 | C3⋊D4 | D12 | C4≀C2 | D18 | C4×D9 | C9⋊D4 | D36 | D12⋊C4 | Dic18⋊C4 |
| kernel | Dic18⋊C4 | C4×Dic9 | C9×M4(2) | D36⋊5C2 | Dic18 | D36 | C3×M4(2) | C36 | C2×C18 | C2×C12 | M4(2) | C12 | C12 | C2×C6 | C9 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 4 | 3 | 6 | 6 | 6 | 2 | 6 |
Matrix representation of Dic18⋊C4 ►in GL4(𝔽73) generated by
| 28 | 70 | 0 | 0 |
| 3 | 31 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [28,3,0,0,70,31,0,0,0,0,27,0,0,0,0,46],[1,72,0,0,0,72,0,0,0,0,0,46,0,0,46,0],[1,72,0,0,0,72,0,0,0,0,46,0,0,0,0,1] >;
Dic18⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{18}\rtimes C_4 % in TeX
G:=Group("Dic18:C4"); // GroupNames label
G:=SmallGroup(288,32);
// by ID
G=gap.SmallGroup(288,32);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=c^4=1,b^2=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b>;
// generators/relations