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), C9⋊1(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
Generators and relations for D18⋊C4
G = < a,b,c | a18=b2=c4=1, bab=a-1, ac=ca, cbc-1=a9b >
Smallest permutation representation of D18⋊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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 36)(34 35)(37 41)(38 40)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(55 59)(56 58)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
(1 62 26 44)(2 63 27 45)(3 64 28 46)(4 65 29 47)(5 66 30 48)(6 67 31 49)(7 68 32 50)(8 69 33 51)(9 70 34 52)(10 71 35 53)(11 72 36 54)(12 55 19 37)(13 56 20 38)(14 57 21 39)(15 58 22 40)(16 59 23 41)(17 60 24 42)(18 61 25 43)
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)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,36)(34,35)(37,41)(38,40)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,62,26,44)(2,63,27,45)(3,64,28,46)(4,65,29,47)(5,66,30,48)(6,67,31,49)(7,68,32,50)(8,69,33,51)(9,70,34,52)(10,71,35,53)(11,72,36,54)(12,55,19,37)(13,56,20,38)(14,57,21,39)(15,58,22,40)(16,59,23,41)(17,60,24,42)(18,61,25,43)>;
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)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,36)(34,35)(37,41)(38,40)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,62,26,44)(2,63,27,45)(3,64,28,46)(4,65,29,47)(5,66,30,48)(6,67,31,49)(7,68,32,50)(8,69,33,51)(9,70,34,52)(10,71,35,53)(11,72,36,54)(12,55,19,37)(13,56,20,38)(14,57,21,39)(15,58,22,40)(16,59,23,41)(17,60,24,42)(18,61,25,43) );
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),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(33,36),(34,35),(37,41),(38,40),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(55,59),(56,58),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)], [(1,62,26,44),(2,63,27,45),(3,64,28,46),(4,65,29,47),(5,66,30,48),(6,67,31,49),(7,68,32,50),(8,69,33,51),(9,70,34,52),(10,71,35,53),(11,72,36,54),(12,55,19,37),(13,56,20,38),(14,57,21,39),(15,58,22,40),(16,59,23,41),(17,60,24,42),(18,61,25,43)]])
D18⋊C4 is a maximal subgroup of
C42⋊2D9 C4×D36 C42⋊7D9 C42⋊3D9 C22⋊C4×D9 Dic9⋊4D4 C22⋊3D36 C23.9D18 D18⋊D4 Dic9.D4 C22.4D36 C4⋊C4⋊7D9 D36⋊C4 D18.D4 C4⋊D36 D18⋊Q8 D18⋊2Q8 C4⋊C4⋊D9 C4×C9⋊D4 C23.28D18 C36⋊7D4 C23⋊2D18 Dic9⋊D4 D18⋊3Q8 C36.23D4 D54⋊C4 D18⋊Dic3 C6.18D36 D18⋊C12 C6.11D36
D18⋊C4 is a maximal quotient of
C42⋊4D9 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 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D9 | C4×S3 | D12 | C3⋊D4 | D18 | C4×D9 | D36 | C9⋊D4 |
| kernel | D18⋊C4 | C2×Dic9 | C2×C36 | C22×D9 | D18 | C2×C12 | C18 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
Matrix representation of D18⋊C4 ►in GL3(𝔽37) generated by
| 1 | 0 | 0 |
| 0 | 6 | 17 |
| 0 | 20 | 26 |
| 36 | 0 | 0 |
| 0 | 6 | 17 |
| 0 | 11 | 31 |
| 6 | 0 | 0 |
| 0 | 5 | 10 |
| 0 | 27 | 32 |
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
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