direct product, non-abelian, supersoluble, monomial
Aliases: C2×He3⋊(C2×C4), C62.11D6, C3⋊Dic3⋊6D6, He3⋊4(C22×C4), C32⋊C12⋊7C22, C6.17(C6.D6), (C2×He3).15C23, C22.11(C32⋊D6), (C22×He3).11C22, C6.89(C2×S32), (C3×C6)⋊2(C4×S3), (C2×C6).56S32, C32⋊3(S3×C2×C4), (C2×He3)⋊3(C2×C4), (C2×C3⋊Dic3)⋊5S3, C2.3(C2×C32⋊D6), (C2×C32⋊C12)⋊8C2, (C2×He3⋊C2)⋊4C4, He3⋊C2⋊3(C2×C4), C3.2(C2×C6.D6), (C3×C6).15(C22×S3), (C22×He3⋊C2).4C2, (C2×He3⋊C2).16C22, SmallGroup(432,321)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×He3⋊(C2×C4) |
Generators and relations for C2×He3⋊(C2×C4)
G = < a,b,c,d,e,f | a2=b3=c3=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, ebe=b-1, bf=fb, cd=dc, ce=ec, fcf-1=c-1, ede=fdf-1=d-1, ef=fe >
Subgroups: 1083 in 221 conjugacy classes, 53 normal (11 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C62, C62, S3×C2×C4, C22×Dic3, He3⋊C2, C2×He3, C2×He3, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C32⋊C12, C2×He3⋊C2, C22×He3, C2×S3×Dic3, He3⋊(C2×C4), C2×C32⋊C12, C22×He3⋊C2, C2×He3⋊(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C6.D6, C2×S32, C32⋊D6, C2×C6.D6, He3⋊(C2×C4), C2×C32⋊D6, C2×He3⋊(C2×C4)
►On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 23)(6 24)(7 21)(8 22)(9 36)(10 33)(11 34)(12 35)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(25 69)(26 70)(27 71)(28 72)(29 68)(30 65)(31 66)(32 67)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)
(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 69 24)(10 70 21)(11 71 22)(12 72 23)(29 64 55)(30 61 56)(31 62 53)(32 63 54)(49 59 66)(50 60 67)(51 57 68)(52 58 65)
(1 19 37)(2 38 20)(3 17 39)(4 40 18)(5 28 35)(6 36 25)(7 26 33)(8 34 27)(9 69 24)(10 21 70)(11 71 22)(12 23 72)(13 41 48)(14 45 42)(15 43 46)(16 47 44)(29 64 55)(30 56 61)(31 62 53)(32 54 63)(49 59 66)(50 67 60)(51 57 68)(52 65 58)
(1 32 24)(2 21 29)(3 30 22)(4 23 31)(5 66 44)(6 41 67)(7 68 42)(8 43 65)(9 19 54)(10 55 20)(11 17 56)(12 53 18)(13 50 25)(14 26 51)(15 52 27)(16 28 49)(33 57 45)(34 46 58)(35 59 47)(36 48 60)(37 63 69)(38 70 64)(39 61 71)(40 72 62)
(1 41)(2 42)(3 43)(4 44)(5 31)(6 32)(7 29)(8 30)(9 60)(10 57)(11 58)(12 59)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(21 68)(22 65)(23 66)(24 67)(25 63)(26 64)(27 61)(28 62)(33 55)(34 56)(35 53)(36 54)(49 72)(50 69)(51 70)(52 71)
(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)| (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,36)(10,33)(11,34)(12,35)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(25,69)(26,70)(27,71)(28,72)(29,68)(30,65)(31,66)(32,67)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,69,24)(10,70,21)(11,71,22)(12,72,23)(29,64,55)(30,61,56)(31,62,53)(32,63,54)(49,59,66)(50,60,67)(51,57,68)(52,58,65), (1,19,37)(2,38,20)(3,17,39)(4,40,18)(5,28,35)(6,36,25)(7,26,33)(8,34,27)(9,69,24)(10,21,70)(11,71,22)(12,23,72)(13,41,48)(14,45,42)(15,43,46)(16,47,44)(29,64,55)(30,56,61)(31,62,53)(32,54,63)(49,59,66)(50,67,60)(51,57,68)(52,65,58), (1,32,24)(2,21,29)(3,30,22)(4,23,31)(5,66,44)(6,41,67)(7,68,42)(8,43,65)(9,19,54)(10,55,20)(11,17,56)(12,53,18)(13,50,25)(14,26,51)(15,52,27)(16,28,49)(33,57,45)(34,46,58)(35,59,47)(36,48,60)(37,63,69)(38,70,64)(39,61,71)(40,72,62), (1,41)(2,42)(3,43)(4,44)(5,31)(6,32)(7,29)(8,30)(9,60)(10,57)(11,58)(12,59)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(21,68)(22,65)(23,66)(24,67)(25,63)(26,64)(27,61)(28,62)(33,55)(34,56)(35,53)(36,54)(49,72)(50,69)(51,70)(52,71), (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( (1,41)(2,42)(3,43)(4,44)(5,23)(6,24)(7,21)(8,22)(9,36)(10,33)(11,34)(12,35)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(25,69)(26,70)(27,71)(28,72)(29,68)(30,65)(31,66)(32,67)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,69,24)(10,70,21)(11,71,22)(12,72,23)(29,64,55)(30,61,56)(31,62,53)(32,63,54)(49,59,66)(50,60,67)(51,57,68)(52,58,65), (1,19,37)(2,38,20)(3,17,39)(4,40,18)(5,28,35)(6,36,25)(7,26,33)(8,34,27)(9,69,24)(10,21,70)(11,71,22)(12,23,72)(13,41,48)(14,45,42)(15,43,46)(16,47,44)(29,64,55)(30,56,61)(31,62,53)(32,54,63)(49,59,66)(50,67,60)(51,57,68)(52,65,58), (1,32,24)(2,21,29)(3,30,22)(4,23,31)(5,66,44)(6,41,67)(7,68,42)(8,43,65)(9,19,54)(10,55,20)(11,17,56)(12,53,18)(13,50,25)(14,26,51)(15,52,27)(16,28,49)(33,57,45)(34,46,58)(35,59,47)(36,48,60)(37,63,69)(38,70,64)(39,61,71)(40,72,62), (1,41)(2,42)(3,43)(4,44)(5,31)(6,32)(7,29)(8,30)(9,60)(10,57)(11,58)(12,59)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(21,68)(22,65)(23,66)(24,67)(25,63)(26,64)(27,61)(28,62)(33,55)(34,56)(35,53)(36,54)(49,72)(50,69)(51,70)(52,71), (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([[(1,41),(2,42),(3,43),(4,44),(5,23),(6,24),(7,21),(8,22),(9,36),(10,33),(11,34),(12,35),(13,37),(14,38),(15,39),(16,40),(17,46),(18,47),(19,48),(20,45),(25,69),(26,70),(27,71),(28,72),(29,68),(30,65),(31,66),(32,67),(49,62),(50,63),(51,64),(52,61),(53,59),(54,60),(55,57),(56,58)], [(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,69,24),(10,70,21),(11,71,22),(12,72,23),(29,64,55),(30,61,56),(31,62,53),(32,63,54),(49,59,66),(50,60,67),(51,57,68),(52,58,65)], [(1,19,37),(2,38,20),(3,17,39),(4,40,18),(5,28,35),(6,36,25),(7,26,33),(8,34,27),(9,69,24),(10,21,70),(11,71,22),(12,23,72),(13,41,48),(14,45,42),(15,43,46),(16,47,44),(29,64,55),(30,56,61),(31,62,53),(32,54,63),(49,59,66),(50,67,60),(51,57,68),(52,65,58)], [(1,32,24),(2,21,29),(3,30,22),(4,23,31),(5,66,44),(6,41,67),(7,68,42),(8,43,65),(9,19,54),(10,55,20),(11,17,56),(12,53,18),(13,50,25),(14,26,51),(15,52,27),(16,28,49),(33,57,45),(34,46,58),(35,59,47),(36,48,60),(37,63,69),(38,70,64),(39,61,71),(40,72,62)], [(1,41),(2,42),(3,43),(4,44),(5,31),(6,32),(7,29),(8,30),(9,60),(10,57),(11,58),(12,59),(13,37),(14,38),(15,39),(16,40),(17,46),(18,47),(19,48),(20,45),(21,68),(22,65),(23,66),(24,67),(25,63),(26,64),(27,61),(28,62),(33,55),(34,56),(35,53),(36,54),(49,72),(50,69),(51,70),(52,71)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 6 | 6 | 12 | 9 | ··· | 9 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | ··· | 18 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4×S3 | S32 | C6.D6 | C2×S32 | C32⋊D6 | He3⋊(C2×C4) | C2×C32⋊D6 |
| kernel | C2×He3⋊(C2×C4) | He3⋊(C2×C4) | C2×C32⋊C12 | C22×He3⋊C2 | C2×He3⋊C2 | C2×C3⋊Dic3 | C3⋊Dic3 | C62 | C3×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of C2×He3⋊(C2×C4) ►in GL10(𝔽13)
| 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 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 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 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 0 | 12 | 12 |
| 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 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 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 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 12 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 3 | 3 | 9 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 | 10 | 10 | 0 | 4 |
| 0 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 10 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 10 | 10 | 3 |
G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,12,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[12,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,12,11,1,1,0,0,0,0,0,0,1,12,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,12,12,0,0,0,0,0,0,12,1,0,0],[0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,10,7,3,10,3,3,0,0,0,0,10,3,3,10,0,0,0,0,0,0,0,0,3,10,3,0,0,0,0,0,0,0,3,10,0,10,0,0,0,0,0,0,9,0,10,10,0,0,0,0,0,0,0,4,3,3] >;
C2×He3⋊(C2×C4) in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes (C_2\times C_4) % in TeX
G:=Group("C2xHe3:(C2xC4)"); // GroupNames label
G:=SmallGroup(432,321);
// by ID
G=gap.SmallGroup(432,321);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^2=f^4=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,d*b*d^-1=b*c^-1,e*b*e=b^-1,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c^-1,e*d*e=f*d*f^-1=d^-1,e*f=f*e>;
// generators/relations