direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C24⋊C9, C62.8A4, C24⋊6(C3×C9), (C23×C6)⋊2C9, (C22×C62).3C3, C32.2(C22⋊A4), (C23×C6).13C32, (C2×C6)⋊(C3.A4), (C2×C6).14(C3×A4), C22⋊2(C3×C3.A4), C3.2(C3×C22⋊A4), SmallGroup(432,553)
Series: Derived ►Chief ►Lower central ►Upper central
| C24 — C3×C24⋊C9 |
Generators and relations for C3×C24⋊C9
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f9=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >
Subgroups: 550 in 190 conjugacy classes, 46 normal (7 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C32, C2×C6, C2×C6, C24, C3×C6, C22×C6, C3×C9, C3.A4, C62, C62, C23×C6, C23×C6, C2×C62, C3×C3.A4, C24⋊C9, C22×C62, C3×C24⋊C9
Quotients: C1, C3, C9, C32, A4, C3×C9, C3.A4, C3×A4, C22⋊A4, C3×C3.A4, C24⋊C9, C3×C22⋊A4, C3×C24⋊C9
►On 108 points
Generators in S108
(1 63 19)(2 55 20)(3 56 21)(4 57 22)(5 58 23)(6 59 24)(7 60 25)(8 61 26)(9 62 27)(10 92 100)(11 93 101)(12 94 102)(13 95 103)(14 96 104)(15 97 105)(16 98 106)(17 99 107)(18 91 108)(28 69 42)(29 70 43)(30 71 44)(31 72 45)(32 64 37)(33 65 38)(34 66 39)(35 67 40)(36 68 41)(46 81 90)(47 73 82)(48 74 83)(49 75 84)(50 76 85)(51 77 86)(52 78 87)(53 79 88)(54 80 89)
(1 106)(2 50)(3 67)(4 100)(5 53)(6 70)(7 103)(8 47)(9 64)(10 57)(11 42)(12 80)(13 60)(14 45)(15 74)(16 63)(17 39)(18 77)(19 98)(20 85)(21 35)(22 92)(23 88)(24 29)(25 95)(26 82)(27 32)(28 93)(30 90)(31 96)(33 84)(34 99)(36 87)(37 62)(38 75)(40 56)(41 78)(43 59)(44 81)(46 71)(48 105)(49 65)(51 108)(52 68)(54 102)(55 76)(58 79)(61 73)(66 107)(69 101)(72 104)(83 97)(86 91)(89 94)
(1 65)(2 107)(3 51)(4 68)(5 101)(6 54)(7 71)(8 104)(9 48)(10 78)(11 58)(12 43)(13 81)(14 61)(15 37)(16 75)(17 55)(18 40)(19 33)(20 99)(21 86)(22 36)(23 93)(24 89)(25 30)(26 96)(27 83)(28 88)(29 94)(31 82)(32 97)(34 85)(35 91)(38 63)(39 76)(41 57)(42 79)(44 60)(45 73)(46 103)(47 72)(49 106)(50 66)(52 100)(53 69)(56 77)(59 80)(62 74)(64 105)(67 108)(70 102)(84 98)(87 92)(90 95)
(1 49)(3 51)(4 52)(6 54)(7 46)(9 48)(10 41)(12 43)(13 44)(15 37)(16 38)(18 40)(19 84)(21 86)(22 87)(24 89)(25 90)(27 83)(29 94)(30 95)(32 97)(33 98)(35 91)(36 92)(56 77)(57 78)(59 80)(60 81)(62 74)(63 75)(64 105)(65 106)(67 108)(68 100)(70 102)(71 103)
(1 49)(2 50)(4 52)(5 53)(7 46)(8 47)(10 41)(11 42)(13 44)(14 45)(16 38)(17 39)(19 84)(20 85)(22 87)(23 88)(25 90)(26 82)(28 93)(30 95)(31 96)(33 98)(34 99)(36 92)(55 76)(57 78)(58 79)(60 81)(61 73)(63 75)(65 106)(66 107)(68 100)(69 101)(71 103)(72 104)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
G:=sub<Sym(108)| (1,63,19)(2,55,20)(3,56,21)(4,57,22)(5,58,23)(6,59,24)(7,60,25)(8,61,26)(9,62,27)(10,92,100)(11,93,101)(12,94,102)(13,95,103)(14,96,104)(15,97,105)(16,98,106)(17,99,107)(18,91,108)(28,69,42)(29,70,43)(30,71,44)(31,72,45)(32,64,37)(33,65,38)(34,66,39)(35,67,40)(36,68,41)(46,81,90)(47,73,82)(48,74,83)(49,75,84)(50,76,85)(51,77,86)(52,78,87)(53,79,88)(54,80,89), (1,106)(2,50)(3,67)(4,100)(5,53)(6,70)(7,103)(8,47)(9,64)(10,57)(11,42)(12,80)(13,60)(14,45)(15,74)(16,63)(17,39)(18,77)(19,98)(20,85)(21,35)(22,92)(23,88)(24,29)(25,95)(26,82)(27,32)(28,93)(30,90)(31,96)(33,84)(34,99)(36,87)(37,62)(38,75)(40,56)(41,78)(43,59)(44,81)(46,71)(48,105)(49,65)(51,108)(52,68)(54,102)(55,76)(58,79)(61,73)(66,107)(69,101)(72,104)(83,97)(86,91)(89,94), (1,65)(2,107)(3,51)(4,68)(5,101)(6,54)(7,71)(8,104)(9,48)(10,78)(11,58)(12,43)(13,81)(14,61)(15,37)(16,75)(17,55)(18,40)(19,33)(20,99)(21,86)(22,36)(23,93)(24,89)(25,30)(26,96)(27,83)(28,88)(29,94)(31,82)(32,97)(34,85)(35,91)(38,63)(39,76)(41,57)(42,79)(44,60)(45,73)(46,103)(47,72)(49,106)(50,66)(52,100)(53,69)(56,77)(59,80)(62,74)(64,105)(67,108)(70,102)(84,98)(87,92)(90,95), (1,49)(3,51)(4,52)(6,54)(7,46)(9,48)(10,41)(12,43)(13,44)(15,37)(16,38)(18,40)(19,84)(21,86)(22,87)(24,89)(25,90)(27,83)(29,94)(30,95)(32,97)(33,98)(35,91)(36,92)(56,77)(57,78)(59,80)(60,81)(62,74)(63,75)(64,105)(65,106)(67,108)(68,100)(70,102)(71,103), (1,49)(2,50)(4,52)(5,53)(7,46)(8,47)(10,41)(11,42)(13,44)(14,45)(16,38)(17,39)(19,84)(20,85)(22,87)(23,88)(25,90)(26,82)(28,93)(30,95)(31,96)(33,98)(34,99)(36,92)(55,76)(57,78)(58,79)(60,81)(61,73)(63,75)(65,106)(66,107)(68,100)(69,101)(71,103)(72,104), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)>;
G:=Group( (1,63,19)(2,55,20)(3,56,21)(4,57,22)(5,58,23)(6,59,24)(7,60,25)(8,61,26)(9,62,27)(10,92,100)(11,93,101)(12,94,102)(13,95,103)(14,96,104)(15,97,105)(16,98,106)(17,99,107)(18,91,108)(28,69,42)(29,70,43)(30,71,44)(31,72,45)(32,64,37)(33,65,38)(34,66,39)(35,67,40)(36,68,41)(46,81,90)(47,73,82)(48,74,83)(49,75,84)(50,76,85)(51,77,86)(52,78,87)(53,79,88)(54,80,89), (1,106)(2,50)(3,67)(4,100)(5,53)(6,70)(7,103)(8,47)(9,64)(10,57)(11,42)(12,80)(13,60)(14,45)(15,74)(16,63)(17,39)(18,77)(19,98)(20,85)(21,35)(22,92)(23,88)(24,29)(25,95)(26,82)(27,32)(28,93)(30,90)(31,96)(33,84)(34,99)(36,87)(37,62)(38,75)(40,56)(41,78)(43,59)(44,81)(46,71)(48,105)(49,65)(51,108)(52,68)(54,102)(55,76)(58,79)(61,73)(66,107)(69,101)(72,104)(83,97)(86,91)(89,94), (1,65)(2,107)(3,51)(4,68)(5,101)(6,54)(7,71)(8,104)(9,48)(10,78)(11,58)(12,43)(13,81)(14,61)(15,37)(16,75)(17,55)(18,40)(19,33)(20,99)(21,86)(22,36)(23,93)(24,89)(25,30)(26,96)(27,83)(28,88)(29,94)(31,82)(32,97)(34,85)(35,91)(38,63)(39,76)(41,57)(42,79)(44,60)(45,73)(46,103)(47,72)(49,106)(50,66)(52,100)(53,69)(56,77)(59,80)(62,74)(64,105)(67,108)(70,102)(84,98)(87,92)(90,95), (1,49)(3,51)(4,52)(6,54)(7,46)(9,48)(10,41)(12,43)(13,44)(15,37)(16,38)(18,40)(19,84)(21,86)(22,87)(24,89)(25,90)(27,83)(29,94)(30,95)(32,97)(33,98)(35,91)(36,92)(56,77)(57,78)(59,80)(60,81)(62,74)(63,75)(64,105)(65,106)(67,108)(68,100)(70,102)(71,103), (1,49)(2,50)(4,52)(5,53)(7,46)(8,47)(10,41)(11,42)(13,44)(14,45)(16,38)(17,39)(19,84)(20,85)(22,87)(23,88)(25,90)(26,82)(28,93)(30,95)(31,96)(33,98)(34,99)(36,92)(55,76)(57,78)(58,79)(60,81)(61,73)(63,75)(65,106)(66,107)(68,100)(69,101)(71,103)(72,104), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108) );
G=PermutationGroup([[(1,63,19),(2,55,20),(3,56,21),(4,57,22),(5,58,23),(6,59,24),(7,60,25),(8,61,26),(9,62,27),(10,92,100),(11,93,101),(12,94,102),(13,95,103),(14,96,104),(15,97,105),(16,98,106),(17,99,107),(18,91,108),(28,69,42),(29,70,43),(30,71,44),(31,72,45),(32,64,37),(33,65,38),(34,66,39),(35,67,40),(36,68,41),(46,81,90),(47,73,82),(48,74,83),(49,75,84),(50,76,85),(51,77,86),(52,78,87),(53,79,88),(54,80,89)], [(1,106),(2,50),(3,67),(4,100),(5,53),(6,70),(7,103),(8,47),(9,64),(10,57),(11,42),(12,80),(13,60),(14,45),(15,74),(16,63),(17,39),(18,77),(19,98),(20,85),(21,35),(22,92),(23,88),(24,29),(25,95),(26,82),(27,32),(28,93),(30,90),(31,96),(33,84),(34,99),(36,87),(37,62),(38,75),(40,56),(41,78),(43,59),(44,81),(46,71),(48,105),(49,65),(51,108),(52,68),(54,102),(55,76),(58,79),(61,73),(66,107),(69,101),(72,104),(83,97),(86,91),(89,94)], [(1,65),(2,107),(3,51),(4,68),(5,101),(6,54),(7,71),(8,104),(9,48),(10,78),(11,58),(12,43),(13,81),(14,61),(15,37),(16,75),(17,55),(18,40),(19,33),(20,99),(21,86),(22,36),(23,93),(24,89),(25,30),(26,96),(27,83),(28,88),(29,94),(31,82),(32,97),(34,85),(35,91),(38,63),(39,76),(41,57),(42,79),(44,60),(45,73),(46,103),(47,72),(49,106),(50,66),(52,100),(53,69),(56,77),(59,80),(62,74),(64,105),(67,108),(70,102),(84,98),(87,92),(90,95)], [(1,49),(3,51),(4,52),(6,54),(7,46),(9,48),(10,41),(12,43),(13,44),(15,37),(16,38),(18,40),(19,84),(21,86),(22,87),(24,89),(25,90),(27,83),(29,94),(30,95),(32,97),(33,98),(35,91),(36,92),(56,77),(57,78),(59,80),(60,81),(62,74),(63,75),(64,105),(65,106),(67,108),(68,100),(70,102),(71,103)], [(1,49),(2,50),(4,52),(5,53),(7,46),(8,47),(10,41),(11,42),(13,44),(14,45),(16,38),(17,39),(19,84),(20,85),(22,87),(23,88),(25,90),(26,82),(28,93),(30,95),(31,96),(33,98),(34,99),(36,92),(55,76),(57,78),(58,79),(60,81),(61,73),(63,75),(65,106),(66,107),(68,100),(69,101),(71,103),(72,104)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2E | 3A | ··· | 3H | 6A | ··· | 6AN | 9A | ··· | 9R |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 16 | ··· | 16 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||
| image | C1 | C3 | C3 | C9 | A4 | C3.A4 | C3×A4 |
| kernel | C3×C24⋊C9 | C24⋊C9 | C22×C62 | C23×C6 | C62 | C2×C6 | C2×C6 |
| # reps | 1 | 6 | 2 | 18 | 5 | 30 | 10 |
Matrix representation of C3×C24⋊C9 ►in GL6(𝔽19)
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 2 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 17 | 6 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 2 | 13 | 17 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 11 | 0 | 0 |
G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[18,0,0,0,0,0,0,1,13,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,2,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1],[18,0,17,0,0,0,0,18,6,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,1,13,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18],[0,2,0,0,0,0,1,13,0,0,0,0,0,17,6,0,0,0,0,0,0,0,0,11,0,0,0,11,0,0,0,0,0,0,11,0] >;
C3×C24⋊C9 in GAP, Magma, Sage, TeX
C_3\times C_2^4\rtimes C_9
% in TeX
G:=Group("C3xC2^4:C9"); // GroupNames label
G:=SmallGroup(432,553);
// by ID
G=gap.SmallGroup(432,553);
# by ID
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,1515,2839,9077,15882]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^9=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,f*c*f^-1=b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations