direct product, non-abelian, supersoluble, monomial
Aliases: C2×C32⋊C6⋊C4, C62.7D6, C3⋊Dic3⋊5D6, He3⋊3(C22×C4), C6.16(S3×Dic3), C32⋊(C22×Dic3), C32⋊C12⋊6C22, He3⋊3C4⋊5C22, (C2×He3).11C23, C22.9(C32⋊D6), (C22×He3).7C22, C6.85(C2×S32), (C3×C6)⋊1(C4×S3), C3⋊S3⋊(C2×Dic3), (C3×C6)⋊(C2×Dic3), C32⋊2(S3×C2×C4), (C2×C3⋊S3).9D6, (C2×C6).53(S32), C3.2(C2×S3×Dic3), (C2×C32⋊C6)⋊3C4, C32⋊C6⋊2(C2×C4), (C2×He3)⋊2(C2×C4), (C2×C3⋊Dic3)⋊3S3, (C2×C3⋊S3)⋊2Dic3, C2.2(C2×C32⋊D6), (C2×C32⋊C12)⋊7C2, (C2×He3⋊3C4)⋊6C2, (C22×C3⋊S3).2S3, (C3×C6).11(C22×S3), (C2×C32⋊C6).9C22, (C22×C32⋊C6).3C2, SmallGroup(432,317)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×C32⋊C6⋊C4 |
Generators and relations for C2×C32⋊C6⋊C4
G = < a,b,c,d,e | a2=b3=c3=d6=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, dbd-1=b-1c-1, dcd-1=ece-1=c-1, ede-1=d-1 >
Subgroups: 1019 in 205 conjugacy classes, 61 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C3 [×3], C4 [×4], C22, C22 [×6], S3 [×8], C6, C6 [×2], C6 [×13], C2×C4 [×6], C23, C32 [×2], C32, Dic3 [×10], C12 [×4], D6 [×12], C2×C6, C2×C6 [×9], C22×C4, C3×S3 [×4], C3⋊S3 [×4], C3×C6 [×2], C3×C6 [×4], C3×C6 [×3], C4×S3 [×8], C2×Dic3 [×9], C2×C12 [×2], C22×S3 [×2], C22×C6, He3, C3×Dic3 [×8], C3⋊Dic3 [×2], S3×C6 [×6], C2×C3⋊S3 [×6], C62 [×2], C62, S3×C2×C4 [×2], C22×Dic3, C32⋊C6 [×4], C2×He3, C2×He3 [×2], S3×Dic3 [×4], C6.D6 [×4], C6×Dic3 [×4], C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C32⋊C12 [×2], He3⋊3C4 [×2], C2×C32⋊C6 [×6], C22×He3, C2×S3×Dic3, C2×C6.D6, C32⋊C6⋊C4 [×4], C2×C32⋊C12, C2×He3⋊3C4, C22×C32⋊C6, C2×C32⋊C6⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C32⋊D6, C2×S3×Dic3, C32⋊C6⋊C4 [×2], C2×C32⋊D6, C2×C32⋊C6⋊C4
►On 72 points
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 65)(8 66)(9 61)(10 62)(11 63)(12 64)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 29 22)(2 30 23)(4 19 26)(5 20 27)(8 68 17)(9 69 18)(11 14 71)(12 15 72)(31 44 40)(32 45 41)(34 37 47)(35 38 48)(50 57 66)(51 58 61)(53 63 60)(54 64 55)
(1 29 22)(2 23 30)(3 25 24)(4 19 26)(5 27 20)(6 21 28)(7 16 67)(8 68 17)(9 18 69)(10 70 13)(11 14 71)(12 72 15)(31 44 40)(32 41 45)(33 46 42)(34 37 47)(35 48 38)(36 39 43)(49 65 56)(50 57 66)(51 61 58)(52 59 62)(53 63 60)(54 55 64)
(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 71 4 68)(2 70 5 67)(3 69 6 72)(7 30 13 20)(8 29 14 19)(9 28 15 24)(10 27 16 23)(11 26 17 22)(12 25 18 21)(31 57 37 63)(32 56 38 62)(33 55 39 61)(34 60 40 66)(35 59 41 65)(36 58 42 64)(43 51 46 54)(44 50 47 53)(45 49 48 52)
G:=sub<Sym(72)| (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,65)(8,66)(9,61)(10,62)(11,63)(12,64)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,29,22)(2,30,23)(4,19,26)(5,20,27)(8,68,17)(9,69,18)(11,14,71)(12,15,72)(31,44,40)(32,45,41)(34,37,47)(35,38,48)(50,57,66)(51,58,61)(53,63,60)(54,64,55), (1,29,22)(2,23,30)(3,25,24)(4,19,26)(5,27,20)(6,21,28)(7,16,67)(8,68,17)(9,18,69)(10,70,13)(11,14,71)(12,72,15)(31,44,40)(32,41,45)(33,46,42)(34,37,47)(35,48,38)(36,39,43)(49,65,56)(50,57,66)(51,61,58)(52,59,62)(53,63,60)(54,55,64), (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), (1,71,4,68)(2,70,5,67)(3,69,6,72)(7,30,13,20)(8,29,14,19)(9,28,15,24)(10,27,16,23)(11,26,17,22)(12,25,18,21)(31,57,37,63)(32,56,38,62)(33,55,39,61)(34,60,40,66)(35,59,41,65)(36,58,42,64)(43,51,46,54)(44,50,47,53)(45,49,48,52)>;
G:=Group( (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,65)(8,66)(9,61)(10,62)(11,63)(12,64)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,40)(20,41)(21,42)(22,37)(23,38)(24,39)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,29,22)(2,30,23)(4,19,26)(5,20,27)(8,68,17)(9,69,18)(11,14,71)(12,15,72)(31,44,40)(32,45,41)(34,37,47)(35,38,48)(50,57,66)(51,58,61)(53,63,60)(54,64,55), (1,29,22)(2,23,30)(3,25,24)(4,19,26)(5,27,20)(6,21,28)(7,16,67)(8,68,17)(9,18,69)(10,70,13)(11,14,71)(12,72,15)(31,44,40)(32,41,45)(33,46,42)(34,37,47)(35,48,38)(36,39,43)(49,65,56)(50,57,66)(51,61,58)(52,59,62)(53,63,60)(54,55,64), (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), (1,71,4,68)(2,70,5,67)(3,69,6,72)(7,30,13,20)(8,29,14,19)(9,28,15,24)(10,27,16,23)(11,26,17,22)(12,25,18,21)(31,57,37,63)(32,56,38,62)(33,55,39,61)(34,60,40,66)(35,59,41,65)(36,58,42,64)(43,51,46,54)(44,50,47,53)(45,49,48,52) );
G=PermutationGroup([(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(7,65),(8,66),(9,61),(10,62),(11,63),(12,64),(13,59),(14,60),(15,55),(16,56),(17,57),(18,58),(19,40),(20,41),(21,42),(22,37),(23,38),(24,39),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(1,29,22),(2,30,23),(4,19,26),(5,20,27),(8,68,17),(9,69,18),(11,14,71),(12,15,72),(31,44,40),(32,45,41),(34,37,47),(35,38,48),(50,57,66),(51,58,61),(53,63,60),(54,64,55)], [(1,29,22),(2,23,30),(3,25,24),(4,19,26),(5,27,20),(6,21,28),(7,16,67),(8,68,17),(9,18,69),(10,70,13),(11,14,71),(12,72,15),(31,44,40),(32,41,45),(33,46,42),(34,37,47),(35,48,38),(36,39,43),(49,65,56),(50,57,66),(51,61,58),(52,59,62),(53,63,60),(54,55,64)], [(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)], [(1,71,4,68),(2,70,5,67),(3,69,6,72),(7,30,13,20),(8,29,14,19),(9,28,15,24),(10,27,16,23),(11,26,17,22),(12,25,18,21),(31,57,37,63),(32,56,38,62),(33,55,39,61),(34,60,40,66),(35,59,41,65),(36,58,42,64),(43,51,46,54),(44,50,47,53),(45,49,48,52)])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | Dic3 | D6 | D6 | C4×S3 | S32 | S3×Dic3 | C2×S32 | C32⋊D6 | C32⋊C6⋊C4 | C2×C32⋊D6 |
| kernel | C2×C32⋊C6⋊C4 | C32⋊C6⋊C4 | C2×C32⋊C12 | C2×He3⋊3C4 | C22×C32⋊C6 | C2×C32⋊C6 | C2×C3⋊Dic3 | C22×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C2×C3⋊S3 | C62 | C3×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of C2×C32⋊C6⋊C4 ►in GL10(𝔽13)
| 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 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 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 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 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 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,12,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,12,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0] >;
C2×C32⋊C6⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes C_6\rtimes C_4
% in TeX
G:=Group("C2xC3^2:C6:C4"); // GroupNames label
G:=SmallGroup(432,317);
// by ID
G=gap.SmallGroup(432,317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^6=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations