direct product, metabelian, supersoluble, monomial
Aliases: C23×C32⋊C6, C62⋊16D6, He3⋊2C24, C62⋊6(C2×C6), (C2×C62)⋊4C6, (C2×C62)⋊6S3, C32⋊(C23×C6), (C23×He3)⋊4C2, (C2×He3)⋊2C23, C32⋊2(S3×C23), (C22×He3)⋊9C22, C3⋊S3⋊(C22×C6), (C3×C6)⋊(C22×C6), C6.50(S3×C2×C6), C3.2(S3×C22×C6), (C22×C3⋊S3)⋊6C6, (C23×C3⋊S3)⋊2C3, (C2×C6).73(S3×C6), (C3×C6)⋊2(C22×S3), (C22×C6).35(C3×S3), (C2×C3⋊S3)⋊6(C2×C6), SmallGroup(432,558)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C23×C32⋊C6 |
Generators and relations for C23×C32⋊C6
G = < a,b,c,d,e,f | a2=b2=c2=d3=e3=f6=1, 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, fdf-1=d-1e-1, fef-1=e-1 >
Subgroups: 2089 in 501 conjugacy classes, 182 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×C6, C22×C6, He3, S3×C6, C2×C3⋊S3, C62, C62, S3×C23, C23×C6, C32⋊C6, C2×He3, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C62, C2×C32⋊C6, C22×He3, S3×C22×C6, C23×C3⋊S3, C22×C32⋊C6, C23×He3, C23×C32⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, S3×C6, S3×C23, C23×C6, C32⋊C6, S3×C2×C6, C2×C32⋊C6, S3×C22×C6, C22×C32⋊C6, C23×C32⋊C6
►On 72 points
Generators in S72
(1 13)(2 14)(3 15)(4 16)(5 8)(6 7)(9 22)(10 21)(11 23)(12 24)(17 19)(18 20)(25 53)(26 54)(27 49)(28 50)(29 51)(30 52)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)(43 71)(44 72)(45 67)(46 68)(47 69)(48 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 20)(2 19)(3 6)(4 5)(7 15)(8 16)(9 10)(11 12)(13 18)(14 17)(21 22)(23 24)(25 65)(26 66)(27 61)(28 62)(29 63)(30 64)(31 71)(32 72)(33 67)(34 68)(35 69)(36 70)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)
(1 4)(2 3)(5 20)(6 19)(7 17)(8 18)(9 24)(10 23)(11 21)(12 22)(13 16)(14 15)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)(37 54)(38 49)(39 50)(40 51)(41 52)(42 53)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(61 71)(62 72)(63 67)(64 68)(65 69)(66 70)
(1 31 70)(2 67 34)(3 63 30)(4 27 66)(5 61 26)(6 29 64)(7 51 58)(8 55 54)(9 59 50)(10 53 56)(11 35 72)(12 69 32)(13 38 48)(14 45 41)(15 57 52)(16 49 60)(17 40 46)(18 43 37)(19 33 68)(20 71 36)(21 25 62)(22 65 28)(23 42 44)(24 47 39)
(1 19 11)(2 12 20)(3 22 5)(4 6 21)(7 10 16)(8 15 9)(13 17 23)(14 24 18)(25 27 29)(26 30 28)(31 33 35)(32 36 34)(37 41 39)(38 40 42)(43 45 47)(44 48 46)(49 51 53)(50 54 52)(55 57 59)(56 60 58)(61 63 65)(62 66 64)(67 69 71)(68 72 70)
(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)
G:=sub<Sym(72)| (1,13)(2,14)(3,15)(4,16)(5,8)(6,7)(9,22)(10,21)(11,23)(12,24)(17,19)(18,20)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,38)(32,39)(33,40)(34,41)(35,42)(36,37)(43,71)(44,72)(45,67)(46,68)(47,69)(48,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,20)(2,19)(3,6)(4,5)(7,15)(8,16)(9,10)(11,12)(13,18)(14,17)(21,22)(23,24)(25,65)(26,66)(27,61)(28,62)(29,63)(30,64)(31,71)(32,72)(33,67)(34,68)(35,69)(36,70)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (1,4)(2,3)(5,20)(6,19)(7,17)(8,18)(9,24)(10,23)(11,21)(12,22)(13,16)(14,15)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(37,54)(38,49)(39,50)(40,51)(41,52)(42,53)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70), (1,31,70)(2,67,34)(3,63,30)(4,27,66)(5,61,26)(6,29,64)(7,51,58)(8,55,54)(9,59,50)(10,53,56)(11,35,72)(12,69,32)(13,38,48)(14,45,41)(15,57,52)(16,49,60)(17,40,46)(18,43,37)(19,33,68)(20,71,36)(21,25,62)(22,65,28)(23,42,44)(24,47,39), (1,19,11)(2,12,20)(3,22,5)(4,6,21)(7,10,16)(8,15,9)(13,17,23)(14,24,18)(25,27,29)(26,30,28)(31,33,35)(32,36,34)(37,41,39)(38,40,42)(43,45,47)(44,48,46)(49,51,53)(50,54,52)(55,57,59)(56,60,58)(61,63,65)(62,66,64)(67,69,71)(68,72,70), (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)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,8)(6,7)(9,22)(10,21)(11,23)(12,24)(17,19)(18,20)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,38)(32,39)(33,40)(34,41)(35,42)(36,37)(43,71)(44,72)(45,67)(46,68)(47,69)(48,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,20)(2,19)(3,6)(4,5)(7,15)(8,16)(9,10)(11,12)(13,18)(14,17)(21,22)(23,24)(25,65)(26,66)(27,61)(28,62)(29,63)(30,64)(31,71)(32,72)(33,67)(34,68)(35,69)(36,70)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (1,4)(2,3)(5,20)(6,19)(7,17)(8,18)(9,24)(10,23)(11,21)(12,22)(13,16)(14,15)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(37,54)(38,49)(39,50)(40,51)(41,52)(42,53)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70), (1,31,70)(2,67,34)(3,63,30)(4,27,66)(5,61,26)(6,29,64)(7,51,58)(8,55,54)(9,59,50)(10,53,56)(11,35,72)(12,69,32)(13,38,48)(14,45,41)(15,57,52)(16,49,60)(17,40,46)(18,43,37)(19,33,68)(20,71,36)(21,25,62)(22,65,28)(23,42,44)(24,47,39), (1,19,11)(2,12,20)(3,22,5)(4,6,21)(7,10,16)(8,15,9)(13,17,23)(14,24,18)(25,27,29)(26,30,28)(31,33,35)(32,36,34)(37,41,39)(38,40,42)(43,45,47)(44,48,46)(49,51,53)(50,54,52)(55,57,59)(56,60,58)(61,63,65)(62,66,64)(67,69,71)(68,72,70), (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) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,8),(6,7),(9,22),(10,21),(11,23),(12,24),(17,19),(18,20),(25,53),(26,54),(27,49),(28,50),(29,51),(30,52),(31,38),(32,39),(33,40),(34,41),(35,42),(36,37),(43,71),(44,72),(45,67),(46,68),(47,69),(48,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66)], [(1,20),(2,19),(3,6),(4,5),(7,15),(8,16),(9,10),(11,12),(13,18),(14,17),(21,22),(23,24),(25,65),(26,66),(27,61),(28,62),(29,63),(30,64),(31,71),(32,72),(33,67),(34,68),(35,69),(36,70),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60)], [(1,4),(2,3),(5,20),(6,19),(7,17),(8,18),(9,24),(10,23),(11,21),(12,22),(13,16),(14,15),(25,35),(26,36),(27,31),(28,32),(29,33),(30,34),(37,54),(38,49),(39,50),(40,51),(41,52),(42,53),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(61,71),(62,72),(63,67),(64,68),(65,69),(66,70)], [(1,31,70),(2,67,34),(3,63,30),(4,27,66),(5,61,26),(6,29,64),(7,51,58),(8,55,54),(9,59,50),(10,53,56),(11,35,72),(12,69,32),(13,38,48),(14,45,41),(15,57,52),(16,49,60),(17,40,46),(18,43,37),(19,33,68),(20,71,36),(21,25,62),(22,65,28),(23,42,44),(24,47,39)], [(1,19,11),(2,12,20),(3,22,5),(4,6,21),(7,10,16),(8,15,9),(13,17,23),(14,24,18),(25,27,29),(26,30,28),(31,33,35),(32,36,34),(37,41,39),(38,40,42),(43,45,47),(44,48,46),(49,51,53),(50,54,52),(55,57,59),(56,60,58),(61,63,65),(62,66,64),(67,69,71),(68,72,70)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 3C | 3D | 3E | 3F | 6A | ··· | 6G | 6H | ··· | 6U | 6V | ··· | 6AP | 6AQ | ··· | 6BF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | C32⋊C6 | C2×C32⋊C6 |
| kernel | C23×C32⋊C6 | C22×C32⋊C6 | C23×He3 | C23×C3⋊S3 | C22×C3⋊S3 | C2×C62 | C2×C62 | C62 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 14 | 1 | 2 | 28 | 2 | 1 | 7 | 2 | 14 | 1 | 7 |
Matrix representation of C23×C32⋊C6 ►in GL10(𝔽7)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(10,GF(7))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6],[6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6],[0,6,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6],[2,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,6,0,0] >;
C23×C32⋊C6 in GAP, Magma, Sage, TeX
C_2^3\times C_3^2\rtimes C_6
% in TeX
G:=Group("C2^3xC3^2:C6"); // GroupNames label
G:=SmallGroup(432,558);
// by ID
G=gap.SmallGroup(432,558);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,4037,537,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^3=f^6=1,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,f*d*f^-1=d^-1*e^-1,f*e*f^-1=e^-1>;
// generators/relations