direct product, metabelian, supersoluble, monomial
Aliases: D9×C3⋊D4, D18⋊7D6, D6⋊3D18, Dic3⋊1D18, C62.71D6, C3⋊5(D4×D9), (C3×D9)⋊2D4, (C2×C6)⋊8D18, (C2×C18)⋊4D6, (S3×C6).7D6, C22⋊3(S3×D9), D6⋊D9⋊6C2, C3⋊D36⋊5C2, (C6×C18)⋊3C22, (Dic3×D9)⋊3C2, (C6×D9)⋊7C22, (C22×D9)⋊4S3, C32.4(S3×D4), (S3×C18)⋊3C22, C9⋊Dic3⋊2C22, (C3×Dic3).7D6, C6.26(C22×D9), C6.D18⋊4C2, C18.26(C22×S3), (C3×C18).26C23, (C9×Dic3)⋊1C22, (C2×C6).6S32, (C2×C6×D9)⋊3C2, (C2×S3×D9)⋊5C2, (C3×C9)⋊8(C2×D4), C6.45(C2×S32), C9⋊2(C2×C3⋊D4), C2.26(C2×S3×D9), (C9×C3⋊D4)⋊3C2, C3.1(S3×C3⋊D4), (C2×C9⋊S3)⋊4C22, (C3×C3⋊D4).3S3, (C3×C6).94(C22×S3), SmallGroup(432,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D9×C3⋊D4
G = < a,b,c,d,e | a9=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1244 in 178 conjugacy classes, 45 normal (41 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C3×D9, C3×D9, S3×C9, C9⋊S3, C3×C18, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, C22×D9, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C2×S32, S3×C2×C6, C9×Dic3, C9⋊Dic3, S3×D9, C6×D9, C6×D9, S3×C18, C2×C9⋊S3, C6×C18, D4×D9, S3×C3⋊D4, Dic3×D9, C3⋊D36, D6⋊D9, C9×C3⋊D4, C6.D18, C2×S3×D9, C2×C6×D9, D9×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, S32, S3×D4, C2×C3⋊D4, C22×D9, C2×S32, S3×D9, D4×D9, S3×C3⋊D4, C2×S3×D9, D9×C3⋊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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 63)(45 62)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 72)(54 71)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,63)(45,62)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,72)(54,71), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)>;
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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,61)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,63)(45,62)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,72)(54,71), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72) );
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)], [(1,21),(2,20),(3,19),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,36),(18,35),(37,61),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,63),(45,62),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,72),(54,71)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,37),(7,38),(8,39),(9,40),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 9A | 9B | 9C | 9D | 9E | 9F | 12 | 18A | 18B | 18C | 18D | ··· | 18O | 18P | 18Q | 18R | 36A | 36B | 36C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 18 | 18 | 18 | 36 | 36 | 36 |
| size | 1 | 1 | 2 | 6 | 9 | 9 | 18 | 54 | 2 | 2 | 4 | 6 | 54 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | D9 | C3⋊D4 | D18 | D18 | D18 | S32 | S3×D4 | C2×S32 | S3×D9 | D4×D9 | S3×C3⋊D4 | C2×S3×D9 | D9×C3⋊D4 |
| kernel | D9×C3⋊D4 | Dic3×D9 | C3⋊D36 | D6⋊D9 | C9×C3⋊D4 | C6.D18 | C2×S3×D9 | C2×C6×D9 | C22×D9 | C3×C3⋊D4 | C3×D9 | D18 | C2×C18 | C3×Dic3 | S3×C6 | C62 | C3⋊D4 | D9 | Dic3 | D6 | C2×C6 | C2×C6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of D9×C3⋊D4 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 31 | 9 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 28 | 1 |
| 0 | 0 | 31 | 9 |
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,33,31,0,0,0,9],[36,0,0,0,0,36,0,0,0,0,28,31,0,0,1,9],[26,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,11,0,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,11,0,0,0,0,0,1,0,0,0,0,1] >;
D9×C3⋊D4 in GAP, Magma, Sage, TeX
D_9\times C_3\rtimes D_4
% in TeX
G:=Group("D9xC3:D4"); // GroupNames label
G:=SmallGroup(432,314);
// by ID
G=gap.SmallGroup(432,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^2=c^3=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations