Aliases: C18.6S4, C9⋊GL2(𝔽3), SL2(𝔽3)⋊D9, Q8⋊C9⋊1S3, (Q8×C9)⋊1S3, C6.2(C3⋊S4), C2.3(C9⋊S4), Q8⋊1(C9⋊S3), C3.1(C6.6S4), (C9×SL2(𝔽3))⋊1C2, (C3×SL2(𝔽3)).5S3, (C3×Q8).2(C3⋊S3), SmallGroup(432,253)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C9×SL2(𝔽3) — C18.6S4 |
| C9×SL2(𝔽3) — C18.6S4 |
Generators and relations for C18.6S4
G = < a,b,c,d,e | a18=d3=e2=1, b2=c2=a9, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a9b, dbd-1=a9bc, ebe=bc, dcd-1=b, ece=a9c, ede=d-1 >
Subgroups: 913 in 76 conjugacy classes, 17 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), D12, C3×Q8, C3×C9, C36, D18, C2×C3⋊S3, Q8⋊2S3, GL2(𝔽3), C9⋊S3, C3×C18, C9⋊C8, Q8⋊C9, D36, Q8×C9, C3×SL2(𝔽3), C2×C9⋊S3, Q8⋊2D9, Q8⋊D9, C6.6S4, C9×SL2(𝔽3), C18.6S4
Quotients: C1, C2, S3, D9, C3⋊S3, S4, GL2(𝔽3), C9⋊S3, C3⋊S4, C6.6S4, C9⋊S4, C18.6S4
►On 72 points
Generators in S72
(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 67 10 58)(2 68 11 59)(3 69 12 60)(4 70 13 61)(5 71 14 62)(6 72 15 63)(7 55 16 64)(8 56 17 65)(9 57 18 66)(19 45 28 54)(20 46 29 37)(21 47 30 38)(22 48 31 39)(23 49 32 40)(24 50 33 41)(25 51 34 42)(26 52 35 43)(27 53 36 44)
(1 21 10 30)(2 22 11 31)(3 23 12 32)(4 24 13 33)(5 25 14 34)(6 26 15 35)(7 27 16 36)(8 28 17 19)(9 29 18 20)(37 57 46 66)(38 58 47 67)(39 59 48 68)(40 60 49 69)(41 61 50 70)(42 62 51 71)(43 63 52 72)(44 64 53 55)(45 65 54 56)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 39 71)(20 40 72)(21 41 55)(22 42 56)(23 43 57)(24 44 58)(25 45 59)(26 46 60)(27 47 61)(28 48 62)(29 49 63)(30 50 64)(31 51 65)(32 52 66)(33 53 67)(34 54 68)(35 37 69)(36 38 70)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 18)(12 17)(13 16)(14 15)(19 23)(20 22)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(37 59)(38 58)(39 57)(40 56)(41 55)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)
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), (1,67,10,58)(2,68,11,59)(3,69,12,60)(4,70,13,61)(5,71,14,62)(6,72,15,63)(7,55,16,64)(8,56,17,65)(9,57,18,66)(19,45,28,54)(20,46,29,37)(21,47,30,38)(22,48,31,39)(23,49,32,40)(24,50,33,41)(25,51,34,42)(26,52,35,43)(27,53,36,44), (1,21,10,30)(2,22,11,31)(3,23,12,32)(4,24,13,33)(5,25,14,34)(6,26,15,35)(7,27,16,36)(8,28,17,19)(9,29,18,20)(37,57,46,66)(38,58,47,67)(39,59,48,68)(40,60,49,69)(41,61,50,70)(42,62,51,71)(43,63,52,72)(44,64,53,55)(45,65,54,56), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,39,71)(20,40,72)(21,41,55)(22,42,56)(23,43,57)(24,44,58)(25,45,59)(26,46,60)(27,47,61)(28,48,62)(29,49,63)(30,50,64)(31,51,65)(32,52,66)(33,53,67)(34,54,68)(35,37,69)(36,38,70), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,15)(19,23)(20,22)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(37,59)(38,58)(39,57)(40,56)(41,55)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)>;
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), (1,67,10,58)(2,68,11,59)(3,69,12,60)(4,70,13,61)(5,71,14,62)(6,72,15,63)(7,55,16,64)(8,56,17,65)(9,57,18,66)(19,45,28,54)(20,46,29,37)(21,47,30,38)(22,48,31,39)(23,49,32,40)(24,50,33,41)(25,51,34,42)(26,52,35,43)(27,53,36,44), (1,21,10,30)(2,22,11,31)(3,23,12,32)(4,24,13,33)(5,25,14,34)(6,26,15,35)(7,27,16,36)(8,28,17,19)(9,29,18,20)(37,57,46,66)(38,58,47,67)(39,59,48,68)(40,60,49,69)(41,61,50,70)(42,62,51,71)(43,63,52,72)(44,64,53,55)(45,65,54,56), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,39,71)(20,40,72)(21,41,55)(22,42,56)(23,43,57)(24,44,58)(25,45,59)(26,46,60)(27,47,61)(28,48,62)(29,49,63)(30,50,64)(31,51,65)(32,52,66)(33,53,67)(34,54,68)(35,37,69)(36,38,70), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,15)(19,23)(20,22)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(37,59)(38,58)(39,57)(40,56)(41,55)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60) );
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)], [(1,67,10,58),(2,68,11,59),(3,69,12,60),(4,70,13,61),(5,71,14,62),(6,72,15,63),(7,55,16,64),(8,56,17,65),(9,57,18,66),(19,45,28,54),(20,46,29,37),(21,47,30,38),(22,48,31,39),(23,49,32,40),(24,50,33,41),(25,51,34,42),(26,52,35,43),(27,53,36,44)], [(1,21,10,30),(2,22,11,31),(3,23,12,32),(4,24,13,33),(5,25,14,34),(6,26,15,35),(7,27,16,36),(8,28,17,19),(9,29,18,20),(37,57,46,66),(38,58,47,67),(39,59,48,68),(40,60,49,69),(41,61,50,70),(42,62,51,71),(43,63,52,72),(44,64,53,55),(45,65,54,56)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,39,71),(20,40,72),(21,41,55),(22,42,56),(23,43,57),(24,44,58),(25,45,59),(26,46,60),(27,47,61),(28,48,62),(29,49,63),(30,50,64),(31,51,65),(32,52,66),(33,53,67),(34,54,68),(35,37,69),(36,38,70)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,18),(12,17),(13,16),(14,15),(19,23),(20,22),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(37,59),(38,58),(39,57),(40,56),(41,55),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 9D | ··· | 9I | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 36A | 36B | 36C |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | ··· | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | 36 | 36 |
| size | 1 | 1 | 108 | 2 | 8 | 8 | 8 | 6 | 2 | 8 | 8 | 8 | 54 | 54 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 2 | 2 | 2 | 8 | ··· | 8 | 12 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | S3 | S3 | S3 | D9 | GL2(𝔽3) | S4 | GL2(𝔽3) | C6.6S4 | C18.6S4 | C3⋊S4 | C9⋊S4 |
| kernel | C18.6S4 | C9×SL2(𝔽3) | Q8⋊C9 | Q8×C9 | C3×SL2(𝔽3) | SL2(𝔽3) | C9 | C18 | C9 | C3 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 9 | 2 | 2 | 1 | 3 | 9 | 1 | 3 |
Matrix representation of C18.6S4 ►in GL4(𝔽73) generated by
| 42 | 45 | 0 | 0 |
| 28 | 70 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 44 |
| 0 | 0 | 17 | 45 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 29 |
| 0 | 0 | 44 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 45 |
| 0 | 0 | 28 | 56 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [42,28,0,0,45,70,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,28,17,0,0,44,45],[1,0,0,0,0,1,0,0,0,0,57,44,0,0,29,16],[1,0,0,0,0,1,0,0,0,0,16,28,0,0,45,56],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C18.6S4 in GAP, Magma, Sage, TeX
C_{18}._6S_4 % in TeX
G:=Group("C18.6S4"); // GroupNames label
G:=SmallGroup(432,253);
// by ID
G=gap.SmallGroup(432,253);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,57,632,142,1011,3784,5681,172,2273,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^18=d^3=e^2=1,b^2=c^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^9*b,d*b*d^-1=a^9*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^9*c,e*d*e=d^-1>;
// generators/relations