metabelian, supersoluble, monomial
Aliases: C12.58S32, (S3×C12)⋊6S3, (S3×C6).42D6, C33⋊7D4⋊8C2, (C3×C12).146D6, C33⋊18(C4○D4), C33⋊12D4⋊6C2, C3⋊Dic3.37D6, C32⋊4Q8⋊14S3, C3⋊4(D6.6D6), (C3×Dic3).43D6, C3⋊1(C12.59D6), C32⋊12(C4○D12), (C32×C6).47C23, C32⋊12(Q8⋊3S3), (C32×C12).48C22, (C32×Dic3).29C22, (S3×C3×C12)⋊7C2, C6.57(C2×S32), C4.7(S3×C3⋊S3), D6.7(C2×C3⋊S3), (C4×S3)⋊2(C3⋊S3), C33⋊8(C2×C4)⋊6C2, C12.26(C2×C3⋊S3), C6.10(C22×C3⋊S3), (S3×C3×C6).26C22, Dic3.12(C2×C3⋊S3), (C3×C32⋊4Q8)⋊11C2, (C3×C6).105(C22×S3), (C3×C3⋊Dic3).20C22, (C2×C33⋊C2).8C22, C2.14(C2×S3×C3⋊S3), SmallGroup(432,669)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.58S32
G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, ac=ca, dad-1=eae=faf=a-1, bc=cb, dbd-1=ebe=fbf=b-1, cd=dc, ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >
Subgroups: 2064 in 304 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2 [×3], C3, C3 [×4], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×19], C6, C6 [×4], C6 [×8], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×8], C12, C12 [×4], C12 [×10], D6, D6 [×18], C2×C6 [×4], C4○D4, C3×S3 [×4], C3⋊S3 [×18], C3×C6, C3×C6 [×4], C3×C6 [×5], Dic6 [×4], C4×S3, C4×S3 [×10], D12 [×11], C3⋊D4 [×8], C2×C12 [×4], C3×Q8, C33, C3×Dic3 [×4], C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12, C3×C12 [×4], C3×C12 [×5], S3×C6 [×4], C2×C3⋊S3 [×18], C62, C4○D12 [×4], Q8⋊3S3, S3×C32, C33⋊C2 [×2], C32×C6, C6.D6 [×8], C3⋊D12 [×8], C3×Dic6 [×4], S3×C12 [×4], C32⋊4Q8, C4×C3⋊S3 [×2], C12⋊S3 [×9], C32⋊7D4 [×2], C6×C12, C32×Dic3, C3×C3⋊Dic3 [×2], C32×C12, S3×C3×C6, C2×C33⋊C2 [×2], D6.6D6 [×4], C12.59D6, C33⋊8(C2×C4) [×2], C33⋊7D4 [×2], S3×C3×C12, C3×C32⋊4Q8, C33⋊12D4, C12.58S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], C23, D6 [×15], C4○D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], C4○D12 [×4], Q8⋊3S3, C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D6.6D6 [×4], C12.59D6, C2×S3×C3⋊S3, C12.58S32
►On 72 points
(1 51 65)(2 66 52)(3 49 67)(4 68 50)(5 30 16)(6 13 31)(7 32 14)(8 15 29)(9 41 22)(10 23 42)(11 43 24)(12 21 44)(17 25 61)(18 62 26)(19 27 63)(20 64 28)(33 58 45)(34 46 59)(35 60 47)(36 48 57)(37 53 70)(38 71 54)(39 55 72)(40 69 56)
(1 26 15)(2 16 27)(3 28 13)(4 14 25)(5 63 66)(6 67 64)(7 61 68)(8 65 62)(9 58 69)(10 70 59)(11 60 71)(12 72 57)(17 50 32)(18 29 51)(19 52 30)(20 31 49)(21 39 36)(22 33 40)(23 37 34)(24 35 38)(41 45 56)(42 53 46)(43 47 54)(44 55 48)
(1 15 26)(2 16 27)(3 13 28)(4 14 25)(5 63 66)(6 64 67)(7 61 68)(8 62 65)(9 69 58)(10 70 59)(11 71 60)(12 72 57)(17 50 32)(18 51 29)(19 52 30)(20 49 31)(21 39 36)(22 40 33)(23 37 34)(24 38 35)(41 56 45)(42 53 46)(43 54 47)(44 55 48)
(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 3)(5 19)(6 18)(7 17)(8 20)(9 45)(10 48)(11 47)(12 46)(13 26)(14 25)(15 28)(16 27)(21 34)(22 33)(23 36)(24 35)(29 64)(30 63)(31 62)(32 61)(37 39)(41 58)(42 57)(43 60)(44 59)(49 65)(50 68)(51 67)(52 66)(53 72)(54 71)(55 70)(56 69)
(1 38)(2 39)(3 40)(4 37)(5 12)(6 9)(7 10)(8 11)(13 22)(14 23)(15 24)(16 21)(17 46)(18 47)(19 48)(20 45)(25 34)(26 35)(27 36)(28 33)(29 43)(30 44)(31 41)(32 42)(49 56)(50 53)(51 54)(52 55)(57 63)(58 64)(59 61)(60 62)(65 71)(66 72)(67 69)(68 70)
G:=sub<Sym(72)| (1,51,65)(2,66,52)(3,49,67)(4,68,50)(5,30,16)(6,13,31)(7,32,14)(8,15,29)(9,41,22)(10,23,42)(11,43,24)(12,21,44)(17,25,61)(18,62,26)(19,27,63)(20,64,28)(33,58,45)(34,46,59)(35,60,47)(36,48,57)(37,53,70)(38,71,54)(39,55,72)(40,69,56), (1,26,15)(2,16,27)(3,28,13)(4,14,25)(5,63,66)(6,67,64)(7,61,68)(8,65,62)(9,58,69)(10,70,59)(11,60,71)(12,72,57)(17,50,32)(18,29,51)(19,52,30)(20,31,49)(21,39,36)(22,33,40)(23,37,34)(24,35,38)(41,45,56)(42,53,46)(43,47,54)(44,55,48), (1,15,26)(2,16,27)(3,13,28)(4,14,25)(5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,69,58)(10,70,59)(11,71,60)(12,72,57)(17,50,32)(18,51,29)(19,52,30)(20,49,31)(21,39,36)(22,40,33)(23,37,34)(24,38,35)(41,56,45)(42,53,46)(43,54,47)(44,55,48), (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), (1,3)(5,19)(6,18)(7,17)(8,20)(9,45)(10,48)(11,47)(12,46)(13,26)(14,25)(15,28)(16,27)(21,34)(22,33)(23,36)(24,35)(29,64)(30,63)(31,62)(32,61)(37,39)(41,58)(42,57)(43,60)(44,59)(49,65)(50,68)(51,67)(52,66)(53,72)(54,71)(55,70)(56,69), (1,38)(2,39)(3,40)(4,37)(5,12)(6,9)(7,10)(8,11)(13,22)(14,23)(15,24)(16,21)(17,46)(18,47)(19,48)(20,45)(25,34)(26,35)(27,36)(28,33)(29,43)(30,44)(31,41)(32,42)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62)(65,71)(66,72)(67,69)(68,70)>;
G:=Group( (1,51,65)(2,66,52)(3,49,67)(4,68,50)(5,30,16)(6,13,31)(7,32,14)(8,15,29)(9,41,22)(10,23,42)(11,43,24)(12,21,44)(17,25,61)(18,62,26)(19,27,63)(20,64,28)(33,58,45)(34,46,59)(35,60,47)(36,48,57)(37,53,70)(38,71,54)(39,55,72)(40,69,56), (1,26,15)(2,16,27)(3,28,13)(4,14,25)(5,63,66)(6,67,64)(7,61,68)(8,65,62)(9,58,69)(10,70,59)(11,60,71)(12,72,57)(17,50,32)(18,29,51)(19,52,30)(20,31,49)(21,39,36)(22,33,40)(23,37,34)(24,35,38)(41,45,56)(42,53,46)(43,47,54)(44,55,48), (1,15,26)(2,16,27)(3,13,28)(4,14,25)(5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,69,58)(10,70,59)(11,71,60)(12,72,57)(17,50,32)(18,51,29)(19,52,30)(20,49,31)(21,39,36)(22,40,33)(23,37,34)(24,38,35)(41,56,45)(42,53,46)(43,54,47)(44,55,48), (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), (1,3)(5,19)(6,18)(7,17)(8,20)(9,45)(10,48)(11,47)(12,46)(13,26)(14,25)(15,28)(16,27)(21,34)(22,33)(23,36)(24,35)(29,64)(30,63)(31,62)(32,61)(37,39)(41,58)(42,57)(43,60)(44,59)(49,65)(50,68)(51,67)(52,66)(53,72)(54,71)(55,70)(56,69), (1,38)(2,39)(3,40)(4,37)(5,12)(6,9)(7,10)(8,11)(13,22)(14,23)(15,24)(16,21)(17,46)(18,47)(19,48)(20,45)(25,34)(26,35)(27,36)(28,33)(29,43)(30,44)(31,41)(32,42)(49,56)(50,53)(51,54)(52,55)(57,63)(58,64)(59,61)(60,62)(65,71)(66,72)(67,69)(68,70) );
G=PermutationGroup([(1,51,65),(2,66,52),(3,49,67),(4,68,50),(5,30,16),(6,13,31),(7,32,14),(8,15,29),(9,41,22),(10,23,42),(11,43,24),(12,21,44),(17,25,61),(18,62,26),(19,27,63),(20,64,28),(33,58,45),(34,46,59),(35,60,47),(36,48,57),(37,53,70),(38,71,54),(39,55,72),(40,69,56)], [(1,26,15),(2,16,27),(3,28,13),(4,14,25),(5,63,66),(6,67,64),(7,61,68),(8,65,62),(9,58,69),(10,70,59),(11,60,71),(12,72,57),(17,50,32),(18,29,51),(19,52,30),(20,31,49),(21,39,36),(22,33,40),(23,37,34),(24,35,38),(41,45,56),(42,53,46),(43,47,54),(44,55,48)], [(1,15,26),(2,16,27),(3,13,28),(4,14,25),(5,63,66),(6,64,67),(7,61,68),(8,62,65),(9,69,58),(10,70,59),(11,71,60),(12,72,57),(17,50,32),(18,51,29),(19,52,30),(20,49,31),(21,39,36),(22,40,33),(23,37,34),(24,38,35),(41,56,45),(42,53,46),(43,54,47),(44,55,48)], [(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)], [(1,3),(5,19),(6,18),(7,17),(8,20),(9,45),(10,48),(11,47),(12,46),(13,26),(14,25),(15,28),(16,27),(21,34),(22,33),(23,36),(24,35),(29,64),(30,63),(31,62),(32,61),(37,39),(41,58),(42,57),(43,60),(44,59),(49,65),(50,68),(51,67),(52,66),(53,72),(54,71),(55,70),(56,69)], [(1,38),(2,39),(3,40),(4,37),(5,12),(6,9),(7,10),(8,11),(13,22),(14,23),(15,24),(16,21),(17,46),(18,47),(19,48),(20,45),(25,34),(26,35),(27,36),(28,33),(29,43),(30,44),(31,41),(32,42),(49,56),(50,53),(51,54),(52,55),(57,63),(58,64),(59,61),(60,62),(65,71),(66,72),(67,69),(68,70)])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | ··· | 6Q | 12A | ··· | 12H | 12I | ··· | 12Q | 12R | ··· | 12Y | 12Z | 12AA |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 3 | 3 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 36 | 36 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | Q8⋊3S3 | C2×S32 | D6.6D6 |
| kernel | C12.58S32 | C33⋊8(C2×C4) | C33⋊7D4 | S3×C3×C12 | C3×C32⋊4Q8 | C33⋊12D4 | S3×C12 | C32⋊4Q8 | C3×Dic3 | C3⋊Dic3 | C3×C12 | S3×C6 | C33 | C32 | C12 | C32 | C6 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 5 | 4 | 2 | 16 | 4 | 1 | 4 | 8 |
Matrix representation of C12.58S32 ►in GL6(𝔽13)
| 1 | 10 | 0 | 0 | 0 | 0 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 4 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 10 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 6 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,10,11,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[6,4,0,0,0,0,7,7,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,10,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,6,0,0,0,0,4,4,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C12.58S32 in GAP, Magma, Sage, TeX
C_{12}._{58}S_3^2 % in TeX
G:=Group("C12.58S3^2"); // GroupNames label
G:=SmallGroup(432,669);
// by ID
G=gap.SmallGroup(432,669);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^4=e^2=f^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=f*a*f=a^-1,b*c=c*b,d*b*d^-1=e*b*e=f*b*f=b^-1,c*d=d*c,e*c*e=f*c*f=c^-1,e*d*e=d^-1,d*f=f*d,f*e*f=d^2*e>;
// generators/relations