metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C18.11D4, C23.2D9, C22⋊2Dic9, C22.7D18, (C2×C18)⋊2C4, C9⋊2(C22⋊C4), C18.9(C2×C4), (C2×C6).23D6, (C2×Dic9)⋊2C2, C2.3(C9⋊D4), (C22×C6).6S3, C2.5(C2×Dic9), (C2×C6).5Dic3, C6.18(C3⋊D4), (C2×C18).7C22, (C22×C18).2C2, C3.(C6.D4), C6.10(C2×Dic3), SmallGroup(144,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18.D4
G = < a,b,c | a18=b4=1, c2=a9, bab-1=cac-1=a-1, cbc-1=a9b-1 >
Smallest permutation representation of C18.D4
►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 23 42 57)(2 22 43 56)(3 21 44 55)(4 20 45 72)(5 19 46 71)(6 36 47 70)(7 35 48 69)(8 34 49 68)(9 33 50 67)(10 32 51 66)(11 31 52 65)(12 30 53 64)(13 29 54 63)(14 28 37 62)(15 27 38 61)(16 26 39 60)(17 25 40 59)(18 24 41 58)
(1 66 10 57)(2 65 11 56)(3 64 12 55)(4 63 13 72)(5 62 14 71)(6 61 15 70)(7 60 16 69)(8 59 17 68)(9 58 18 67)(19 46 28 37)(20 45 29 54)(21 44 30 53)(22 43 31 52)(23 42 32 51)(24 41 33 50)(25 40 34 49)(26 39 35 48)(27 38 36 47)
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), (1,23,42,57)(2,22,43,56)(3,21,44,55)(4,20,45,72)(5,19,46,71)(6,36,47,70)(7,35,48,69)(8,34,49,68)(9,33,50,67)(10,32,51,66)(11,31,52,65)(12,30,53,64)(13,29,54,63)(14,28,37,62)(15,27,38,61)(16,26,39,60)(17,25,40,59)(18,24,41,58), (1,66,10,57)(2,65,11,56)(3,64,12,55)(4,63,13,72)(5,62,14,71)(6,61,15,70)(7,60,16,69)(8,59,17,68)(9,58,18,67)(19,46,28,37)(20,45,29,54)(21,44,30,53)(22,43,31,52)(23,42,32,51)(24,41,33,50)(25,40,34,49)(26,39,35,48)(27,38,36,47)>;
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), (1,23,42,57)(2,22,43,56)(3,21,44,55)(4,20,45,72)(5,19,46,71)(6,36,47,70)(7,35,48,69)(8,34,49,68)(9,33,50,67)(10,32,51,66)(11,31,52,65)(12,30,53,64)(13,29,54,63)(14,28,37,62)(15,27,38,61)(16,26,39,60)(17,25,40,59)(18,24,41,58), (1,66,10,57)(2,65,11,56)(3,64,12,55)(4,63,13,72)(5,62,14,71)(6,61,15,70)(7,60,16,69)(8,59,17,68)(9,58,18,67)(19,46,28,37)(20,45,29,54)(21,44,30,53)(22,43,31,52)(23,42,32,51)(24,41,33,50)(25,40,34,49)(26,39,35,48)(27,38,36,47) );
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)], [(1,23,42,57),(2,22,43,56),(3,21,44,55),(4,20,45,72),(5,19,46,71),(6,36,47,70),(7,35,48,69),(8,34,49,68),(9,33,50,67),(10,32,51,66),(11,31,52,65),(12,30,53,64),(13,29,54,63),(14,28,37,62),(15,27,38,61),(16,26,39,60),(17,25,40,59),(18,24,41,58)], [(1,66,10,57),(2,65,11,56),(3,64,12,55),(4,63,13,72),(5,62,14,71),(6,61,15,70),(7,60,16,69),(8,59,17,68),(9,58,18,67),(19,46,28,37),(20,45,29,54),(21,44,30,53),(22,43,31,52),(23,42,32,51),(24,41,33,50),(25,40,34,49),(26,39,35,48),(27,38,36,47)]])
C18.D4 is a maximal subgroup of
C22.D36 C23⋊2Dic9 C23.16D18 C22⋊2Dic18 C23.8D18 C22⋊C4×D9 C23.9D18 Dic9.D4 C36.49D4 C23.26D18 C4×C9⋊D4 C23.28D18 D4×Dic9 C23.23D18 C36.17D4 C23⋊2D18 C36⋊2D4 Dic9⋊D4 C24⋊4D9 C54.D4 C18.S4 D6⋊Dic9 C62.27D6 C62.127D6 A4⋊Dic9
C18.D4 is a maximal quotient of
C36.55D4 C18.C42 C36.D4 D4⋊Dic9 C23⋊2Dic9 C36.9D4 Q8⋊2Dic9 Q8⋊3Dic9 C54.D4 D6⋊Dic9 C62.127D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | ··· | 6G | 9A | 9B | 9C | 18A | ··· | 18U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | D9 | C3⋊D4 | Dic9 | D18 | C9⋊D4 |
| kernel | C18.D4 | C2×Dic9 | C22×C18 | C2×C18 | C22×C6 | C18 | C2×C6 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 |
Matrix representation of C18.D4 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 25 |
| 31 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 6 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 36 | 0 |
G:=sub<GL(3,GF(37))| [36,0,0,0,3,0,0,0,25],[31,0,0,0,0,1,0,1,0],[6,0,0,0,0,36,0,1,0] >;
C18.D4 in GAP, Magma, Sage, TeX
C_{18}.D_4 % in TeX
G:=Group("C18.D4"); // GroupNames label
G:=SmallGroup(144,19);
// by ID
G=gap.SmallGroup(144,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^18=b^4=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations
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