direct product, non-abelian, soluble, monomial
Aliases: C6×A4⋊C4, (C2×A4)⋊C12, (C6×A4)⋊1C4, C2.2(C6×S4), C24.(C3×S3), A4⋊2(C2×C12), C6.48(C2×S4), (C2×C6).21S4, (C22×A4).C6, C23⋊(C3×Dic3), C22⋊(C6×Dic3), C22.6(C3×S4), C23.4(S3×C6), (C23×C6).1S3, (C22×C6)⋊1Dic3, (C22×C6).11D6, (C6×A4).11C22, (A4×C2×C6).1C2, (C3×A4)⋊7(C2×C4), (C2×A4).4(C2×C6), (C2×C6)⋊2(C2×Dic3), SmallGroup(288,905)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C6×A4⋊C4 |
Generators and relations for C6×A4⋊C4
G = < a,b,c,d,e | a6=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Subgroups: 426 in 136 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C3 [×2], C4 [×4], C22 [×2], C22 [×10], C6, C6 [×2], C6 [×10], C2×C4 [×8], C23, C23 [×2], C23 [×4], C32, Dic3 [×2], C12 [×4], A4, A4, C2×C6 [×2], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C3×C6 [×3], C2×Dic3, C2×C12 [×8], C2×A4, C2×A4 [×2], C2×A4 [×3], C22×C6, C22×C6 [×2], C22×C6 [×4], C2×C22⋊C4, C3×Dic3 [×2], C3×A4, C62, C3×C22⋊C4 [×4], A4⋊C4 [×2], C22×C12 [×2], C22×A4, C22×A4, C23×C6, C6×Dic3, C6×A4, C6×A4 [×2], C6×C22⋊C4, C2×A4⋊C4, C3×A4⋊C4 [×2], A4×C2×C6, C6×A4⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C3×S3, C2×Dic3, C2×C12, S4, C3×Dic3 [×2], S3×C6, A4⋊C4 [×2], C2×S4, C6×Dic3, C3×S4, C2×A4⋊C4, C3×A4⋊C4 [×2], C6×S4, C6×A4⋊C4
►On 72 points
Generators in S72
(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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 44)(14 45)(15 46)(16 47)(17 48)(18 43)(19 22)(20 23)(21 24)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(67 70)(68 71)(69 72)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 64)(26 65)(27 66)(28 61)(29 62)(30 63)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)
(1 49 10)(2 50 11)(3 51 12)(4 52 7)(5 53 8)(6 54 9)(13 63 71)(14 64 72)(15 65 67)(16 66 68)(17 61 69)(18 62 70)(19 42 31)(20 37 32)(21 38 33)(22 39 34)(23 40 35)(24 41 36)(25 60 45)(26 55 46)(27 56 47)(28 57 48)(29 58 43)(30 59 44)
(1 67 31 55)(2 68 32 56)(3 69 33 57)(4 70 34 58)(5 71 35 59)(6 72 36 60)(7 18 39 43)(8 13 40 44)(9 14 41 45)(10 15 42 46)(11 16 37 47)(12 17 38 48)(19 26 49 65)(20 27 50 66)(21 28 51 61)(22 29 52 62)(23 30 53 63)(24 25 54 64)
G:=sub<Sym(72)| (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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,44)(14,45)(15,46)(16,47)(17,48)(18,43)(19,22)(20,23)(21,24)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69), (1,49,10)(2,50,11)(3,51,12)(4,52,7)(5,53,8)(6,54,9)(13,63,71)(14,64,72)(15,65,67)(16,66,68)(17,61,69)(18,62,70)(19,42,31)(20,37,32)(21,38,33)(22,39,34)(23,40,35)(24,41,36)(25,60,45)(26,55,46)(27,56,47)(28,57,48)(29,58,43)(30,59,44), (1,67,31,55)(2,68,32,56)(3,69,33,57)(4,70,34,58)(5,71,35,59)(6,72,36,60)(7,18,39,43)(8,13,40,44)(9,14,41,45)(10,15,42,46)(11,16,37,47)(12,17,38,48)(19,26,49,65)(20,27,50,66)(21,28,51,61)(22,29,52,62)(23,30,53,63)(24,25,54,64)>;
G:=Group( (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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,44)(14,45)(15,46)(16,47)(17,48)(18,43)(19,22)(20,23)(21,24)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69), (1,49,10)(2,50,11)(3,51,12)(4,52,7)(5,53,8)(6,54,9)(13,63,71)(14,64,72)(15,65,67)(16,66,68)(17,61,69)(18,62,70)(19,42,31)(20,37,32)(21,38,33)(22,39,34)(23,40,35)(24,41,36)(25,60,45)(26,55,46)(27,56,47)(28,57,48)(29,58,43)(30,59,44), (1,67,31,55)(2,68,32,56)(3,69,33,57)(4,70,34,58)(5,71,35,59)(6,72,36,60)(7,18,39,43)(8,13,40,44)(9,14,41,45)(10,15,42,46)(11,16,37,47)(12,17,38,48)(19,26,49,65)(20,27,50,66)(21,28,51,61)(22,29,52,62)(23,30,53,63)(24,25,54,64) );
G=PermutationGroup([(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)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,44),(14,45),(15,46),(16,47),(17,48),(18,43),(19,22),(20,23),(21,24),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(49,52),(50,53),(51,54),(55,58),(56,59),(57,60),(67,70),(68,71),(69,72)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,64),(26,65),(27,66),(28,61),(29,62),(30,63),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69)], [(1,49,10),(2,50,11),(3,51,12),(4,52,7),(5,53,8),(6,54,9),(13,63,71),(14,64,72),(15,65,67),(16,66,68),(17,61,69),(18,62,70),(19,42,31),(20,37,32),(21,38,33),(22,39,34),(23,40,35),(24,41,36),(25,60,45),(26,55,46),(27,56,47),(28,57,48),(29,58,43),(30,59,44)], [(1,67,31,55),(2,68,32,56),(3,69,33,57),(4,70,34,58),(5,71,35,59),(6,72,36,60),(7,18,39,43),(8,13,40,44),(9,14,41,45),(10,15,42,46),(11,16,37,47),(12,17,38,48),(19,26,49,65),(20,27,50,66),(21,28,51,61),(22,29,52,62),(23,30,53,63),(24,25,54,64)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6F | 6G | ··· | 6N | 6O | ··· | 6W | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 8 | 8 | 8 | 6 | ··· | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 8 | ··· | 8 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | - | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 | S4 | A4⋊C4 | C2×S4 | C3×S4 | C3×A4⋊C4 | C6×S4 |
| kernel | C6×A4⋊C4 | C3×A4⋊C4 | A4×C2×C6 | C2×A4⋊C4 | C6×A4 | A4⋊C4 | C22×A4 | C2×A4 | C23×C6 | C22×C6 | C22×C6 | C24 | C23 | C23 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C6×A4⋊C4 ►in GL5(𝔽13)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 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 |
| 3 | 0 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 8 | 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[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],[3,9,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[8,0,0,0,0,1,5,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0] >;
C6×A4⋊C4 in GAP, Magma, Sage, TeX
C_6\times A_4\rtimes C_4
% in TeX
G:=Group("C6xA4:C4"); // GroupNames label
G:=SmallGroup(288,905);
// by ID
G=gap.SmallGroup(288,905);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,84,1684,6053,285,3534,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations