direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×D9, C36⋊C23, C23⋊4D18, D18⋊2C23, D36⋊7C22, C18.5C24, Dic9⋊1C23, (C2×C4)⋊6D18, C18⋊2(C2×D4), (C2×C18)⋊C23, C9⋊2(C22×D4), (D4×C18)⋊5C2, C4⋊1(C22×D9), (C2×D36)⋊11C2, (C2×C36)⋊2C22, (C3×D4).35D6, (C6×D4).12S3, C6.104(S3×D4), (C2×C12).98D6, (C23×D9)⋊4C2, (D4×C9)⋊5C22, (C4×D9)⋊3C22, C9⋊D4⋊1C22, C2.6(C23×D9), C6.42(S3×C23), (C22×C6).58D6, C22⋊2(C22×D9), C12.59(C22×S3), (C22×C18)⋊4C22, (C2×Dic9)⋊8C22, (C22×D9)⋊6C22, C3.(C2×S3×D4), (C2×C4×D9)⋊3C2, (C2×C9⋊D4)⋊9C2, (C2×C6).7(C22×S3), SmallGroup(288,356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×D9
G = < a,b,c,d,e | a2=b4=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1704 in 354 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C24, D9, D9, C18, C18, C18, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×D4, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×D9, C22×D9, C22×C18, C2×S3×D4, C2×C4×D9, C2×D36, D4×D9, C2×C9⋊D4, D4×C18, C23×D9, C2×D4×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D9, C22×S3, C22×D4, D18, S3×D4, S3×C23, C22×D9, C2×S3×D4, D4×D9, C23×D9, C2×D4×D9
►On 72 points
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 68 14 59)(2 69 15 60)(3 70 16 61)(4 71 17 62)(5 72 18 63)(6 64 10 55)(7 65 11 56)(8 66 12 57)(9 67 13 58)(19 46 28 37)(20 47 29 38)(21 48 30 39)(22 49 31 40)(23 50 32 41)(24 51 33 42)(25 52 34 43)(26 53 35 44)(27 54 36 45)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)
G:=sub<Sym(72)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,68,14,59)(2,69,15,60)(3,70,16,61)(4,71,17,62)(5,72,18,63)(6,64,10,55)(7,65,11,56)(8,66,12,57)(9,67,13,58)(19,46,28,37)(20,47,29,38)(21,48,30,39)(22,49,31,40)(23,50,32,41)(24,51,33,42)(25,52,34,43)(26,53,35,44)(27,54,36,45), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,68,14,59)(2,69,15,60)(3,70,16,61)(4,71,17,62)(5,72,18,63)(6,64,10,55)(7,65,11,56)(8,66,12,57)(9,67,13,58)(19,46,28,37)(20,47,29,38)(21,48,30,39)(22,49,31,40)(23,50,32,41)(24,51,33,42)(25,52,34,43)(26,53,35,44)(27,54,36,45), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68) );
G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(1,68,14,59),(2,69,15,60),(3,70,16,61),(4,71,17,62),(5,72,18,63),(6,64,10,55),(7,65,11,56),(8,66,12,57),(9,67,13,58),(19,46,28,37),(20,47,29,38),(21,48,30,39),(22,49,31,40),(23,50,32,41),(24,51,33,42),(25,52,34,43),(26,53,35,44),(27,54,36,45)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,55),(7,56),(8,57),(9,58),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,62),(56,61),(57,60),(58,59),(64,71),(65,70),(66,69),(67,68)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D9 | D18 | D18 | D18 | S3×D4 | D4×D9 |
| kernel | C2×D4×D9 | C2×C4×D9 | C2×D36 | D4×D9 | C2×C9⋊D4 | D4×C18 | C23×D9 | C6×D4 | D18 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 4 | 1 | 4 | 2 | 3 | 3 | 12 | 6 | 2 | 6 |
Matrix representation of C2×D4×D9 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 13 |
| 0 | 0 | 0 | 0 | 34 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 13 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 6 | 0 | 0 |
| 0 | 0 | 31 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 26 | 0 | 0 |
| 0 | 0 | 20 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,34,0,0,0,0,13,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,13,36],[0,36,0,0,0,0,1,36,0,0,0,0,0,0,20,31,0,0,0,0,6,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,20,0,0,0,0,26,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×D4×D9 in GAP, Magma, Sage, TeX
C_2\times D_4\times D_9
% in TeX
G:=Group("C2xD4xD9"); // GroupNames label
G:=SmallGroup(288,356);
// by ID
G=gap.SmallGroup(288,356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,185,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations