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