direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.32D6, C62.13Q8, C62.106D4, (C6×C12)⋊9C4, (C2×C12)⋊3C12, C6.5(C4×C12), (C6×Dic3)⋊4C4, (C2×C12)⋊3Dic3, (C2×C6).72D12, C6.52(D6⋊C4), (C2×Dic3)⋊2C12, C62.96(C2×C4), C23.37(S3×C6), (C3×C6).15C42, C6.21(C4×Dic3), C2.5(Dic3×C12), (C2×C6).22Dic6, C22.12(S3×C12), (C22×C12).15C6, (C22×C12).14S3, C6.21(C4⋊Dic3), C22.11(C3×D12), (C22×C6).169D6, C22.3(C3×Dic6), C6.25(Dic3⋊C4), (C2×C62).91C22, (C22×Dic3).2C6, C22.11(C6×Dic3), C6.30(C6.D4), C32⋊5(C2.C42), C6.9(C3×C4⋊C4), (C2×C6×C12).1C2, C2.2(C3×D6⋊C4), (C2×C6).7(C3×Q8), (C2×C6).82(C4×S3), (C2×C4)⋊2(C3×Dic3), (C2×C6).41(C3×D4), C3⋊(C3×C2.C42), C2.2(C3×C4⋊Dic3), (Dic3×C2×C6).2C2, (C3×C6).35(C4⋊C4), (C2×C6).16(C2×C12), C6.11(C3×C22⋊C4), C2.2(C3×Dic3⋊C4), (C22×C4).6(C3×S3), (C22×C6).56(C2×C6), (C2×C6).60(C2×Dic3), C2.2(C3×C6.D4), C22.16(C3×C3⋊D4), (C2×C6).109(C3⋊D4), (C3×C6).63(C22⋊C4), SmallGroup(288,265)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.32D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >
Subgroups: 346 in 179 conjugacy classes, 90 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×6], C6 [×8], C6 [×7], C2×C4 [×2], C2×C4 [×10], C23, C32, Dic3 [×4], C12 [×12], C2×C6 [×6], C2×C6 [×8], C2×C6 [×7], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×18], C22×C6 [×2], C22×C6, C2.C42, C3×Dic3 [×4], C3×C12 [×2], C62 [×3], C62 [×4], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C6×C12 [×2], C6×C12 [×2], C2×C62, C23.32D6, C3×C2.C42, Dic3×C2×C6 [×2], C2×C6×C12, C3×C23.32D6
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], C12 [×6], D6, C2×C6, C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×S3, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C2.C42, C3×Dic3 [×2], S3×C6, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C3×Dic6, S3×C12 [×2], C3×D12, C6×Dic3, C3×C3⋊D4 [×2], C23.32D6, C3×C2.C42, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×D6⋊C4 [×2], C3×C6.D4, C3×C23.32D6
►On 96 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 73)(60 74)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(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)
(1 69 91 56)(2 81 92 24)(3 67 93 54)(4 79 94 22)(5 65 95 52)(6 77 96 20)(7 63 85 50)(8 75 86 18)(9 61 87 60)(10 73 88 16)(11 71 89 58)(12 83 90 14)(13 33 82 46)(15 31 84 44)(17 29 74 42)(19 27 76 40)(21 25 78 38)(23 35 80 48)(26 64 39 51)(28 62 41 49)(30 72 43 59)(32 70 45 57)(34 68 47 55)(36 66 37 53)
G:=sub<Sym(96)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,69)(14,70)(15,71)(16,72)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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), (1,69,91,56)(2,81,92,24)(3,67,93,54)(4,79,94,22)(5,65,95,52)(6,77,96,20)(7,63,85,50)(8,75,86,18)(9,61,87,60)(10,73,88,16)(11,71,89,58)(12,83,90,14)(13,33,82,46)(15,31,84,44)(17,29,74,42)(19,27,76,40)(21,25,78,38)(23,35,80,48)(26,64,39,51)(28,62,41,49)(30,72,43,59)(32,70,45,57)(34,68,47,55)(36,66,37,53)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,69)(14,70)(15,71)(16,72)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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), (1,69,91,56)(2,81,92,24)(3,67,93,54)(4,79,94,22)(5,65,95,52)(6,77,96,20)(7,63,85,50)(8,75,86,18)(9,61,87,60)(10,73,88,16)(11,71,89,58)(12,83,90,14)(13,33,82,46)(15,31,84,44)(17,29,74,42)(19,27,76,40)(21,25,78,38)(23,35,80,48)(26,64,39,51)(28,62,41,49)(30,72,43,59)(32,70,45,57)(34,68,47,55)(36,66,37,53) );
G=PermutationGroup([(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,69),(14,70),(15,71),(16,72),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,95),(26,96),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,73),(60,74)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(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)], [(1,69,91,56),(2,81,92,24),(3,67,93,54),(4,79,94,22),(5,65,95,52),(6,77,96,20),(7,63,85,50),(8,75,86,18),(9,61,87,60),(10,73,88,16),(11,71,89,58),(12,83,90,14),(13,33,82,46),(15,31,84,44),(17,29,74,42),(19,27,76,40),(21,25,78,38),(23,35,80,48),(26,64,39,51),(28,62,41,49),(30,72,43,59),(32,70,45,57),(34,68,47,55),(36,66,37,53)])
108 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6N | 6O | ··· | 6AI | 12A | ··· | 12AF | 12AG | ··· | 12AV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | C3×Dic6 | S3×C12 | C3×D12 | C3×C3⋊D4 |
| kernel | C3×C23.32D6 | Dic3×C2×C6 | C2×C6×C12 | C23.32D6 | C6×Dic3 | C6×C12 | C22×Dic3 | C22×C12 | C2×Dic3 | C2×C12 | C22×C12 | C62 | C62 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 6 | 2 | 4 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C3×C23.32D6 ►in GL5(𝔽13)
| 9 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 12 | 2 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 10 |
| 0 | 0 | 0 | 6 | 11 |
G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,7,12,0,0,0,0,2],[5,0,0,0,0,0,0,6,0,0,0,2,0,0,0,0,0,0,2,6,0,0,0,10,11] >;
C3×C23.32D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{32}D_6 % in TeX
G:=Group("C3xC2^3.32D6"); // GroupNames label
G:=SmallGroup(288,265);
// by ID
G=gap.SmallGroup(288,265);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations