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, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C32, Dic3, C12, A4, A4, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C2×Dic3, C2×C12, C2×A4, C2×A4, C2×A4, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C3×A4, C62, C3×C22⋊C4, A4⋊C4, C22×C12, C22×A4, C22×A4, C23×C6, C6×Dic3, C6×A4, C6×A4, C6×C22⋊C4, C2×A4⋊C4, C3×A4⋊C4, A4×C2×C6, C6×A4⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C3×S3, C2×Dic3, C2×C12, S4, C3×Dic3, S3×C6, A4⋊C4, C2×S4, C6×Dic3, C3×S4, C2×A4⋊C4, C3×A4⋊C4, 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 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 47)(14 48)(15 43)(16 44)(17 45)(18 46)(19 22)(20 23)(21 24)(25 66)(26 61)(27 62)(28 63)(29 64)(30 65)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(67 70)(68 71)(69 72)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 52)(20 53)(21 54)(22 49)(23 50)(24 51)(25 63)(26 64)(27 65)(28 66)(29 61)(30 62)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(55 68)(56 69)(57 70)(58 71)(59 72)(60 67)
(1 54 7)(2 49 8)(3 50 9)(4 51 10)(5 52 11)(6 53 12)(13 63 68)(14 64 69)(15 65 70)(16 66 71)(17 61 72)(18 62 67)(19 40 32)(20 41 33)(21 42 34)(22 37 35)(23 38 36)(24 39 31)(25 58 47)(26 59 48)(27 60 43)(28 55 44)(29 56 45)(30 57 46)
(1 70 34 60)(2 71 35 55)(3 72 36 56)(4 67 31 57)(5 68 32 58)(6 69 33 59)(7 15 42 43)(8 16 37 44)(9 17 38 45)(10 18 39 46)(11 13 40 47)(12 14 41 48)(19 25 52 63)(20 26 53 64)(21 27 54 65)(22 28 49 66)(23 29 50 61)(24 30 51 62)
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), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,47)(14,48)(15,43)(16,44)(17,45)(18,46)(19,22)(20,23)(21,24)(25,66)(26,61)(27,62)(28,63)(29,64)(30,65)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,52)(20,53)(21,54)(22,49)(23,50)(24,51)(25,63)(26,64)(27,65)(28,66)(29,61)(30,62)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(55,68)(56,69)(57,70)(58,71)(59,72)(60,67), (1,54,7)(2,49,8)(3,50,9)(4,51,10)(5,52,11)(6,53,12)(13,63,68)(14,64,69)(15,65,70)(16,66,71)(17,61,72)(18,62,67)(19,40,32)(20,41,33)(21,42,34)(22,37,35)(23,38,36)(24,39,31)(25,58,47)(26,59,48)(27,60,43)(28,55,44)(29,56,45)(30,57,46), (1,70,34,60)(2,71,35,55)(3,72,36,56)(4,67,31,57)(5,68,32,58)(6,69,33,59)(7,15,42,43)(8,16,37,44)(9,17,38,45)(10,18,39,46)(11,13,40,47)(12,14,41,48)(19,25,52,63)(20,26,53,64)(21,27,54,65)(22,28,49,66)(23,29,50,61)(24,30,51,62)>;
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), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,39)(8,40)(9,41)(10,42)(11,37)(12,38)(13,47)(14,48)(15,43)(16,44)(17,45)(18,46)(19,22)(20,23)(21,24)(25,66)(26,61)(27,62)(28,63)(29,64)(30,65)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,52)(20,53)(21,54)(22,49)(23,50)(24,51)(25,63)(26,64)(27,65)(28,66)(29,61)(30,62)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(55,68)(56,69)(57,70)(58,71)(59,72)(60,67), (1,54,7)(2,49,8)(3,50,9)(4,51,10)(5,52,11)(6,53,12)(13,63,68)(14,64,69)(15,65,70)(16,66,71)(17,61,72)(18,62,67)(19,40,32)(20,41,33)(21,42,34)(22,37,35)(23,38,36)(24,39,31)(25,58,47)(26,59,48)(27,60,43)(28,55,44)(29,56,45)(30,57,46), (1,70,34,60)(2,71,35,55)(3,72,36,56)(4,67,31,57)(5,68,32,58)(6,69,33,59)(7,15,42,43)(8,16,37,44)(9,17,38,45)(10,18,39,46)(11,13,40,47)(12,14,41,48)(19,25,52,63)(20,26,53,64)(21,27,54,65)(22,28,49,66)(23,29,50,61)(24,30,51,62) );
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)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,39),(8,40),(9,41),(10,42),(11,37),(12,38),(13,47),(14,48),(15,43),(16,44),(17,45),(18,46),(19,22),(20,23),(21,24),(25,66),(26,61),(27,62),(28,63),(29,64),(30,65),(49,52),(50,53),(51,54),(55,58),(56,59),(57,60),(67,70),(68,71),(69,72)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,52),(20,53),(21,54),(22,49),(23,50),(24,51),(25,63),(26,64),(27,65),(28,66),(29,61),(30,62),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(55,68),(56,69),(57,70),(58,71),(59,72),(60,67)], [(1,54,7),(2,49,8),(3,50,9),(4,51,10),(5,52,11),(6,53,12),(13,63,68),(14,64,69),(15,65,70),(16,66,71),(17,61,72),(18,62,67),(19,40,32),(20,41,33),(21,42,34),(22,37,35),(23,38,36),(24,39,31),(25,58,47),(26,59,48),(27,60,43),(28,55,44),(29,56,45),(30,57,46)], [(1,70,34,60),(2,71,35,55),(3,72,36,56),(4,67,31,57),(5,68,32,58),(6,69,33,59),(7,15,42,43),(8,16,37,44),(9,17,38,45),(10,18,39,46),(11,13,40,47),(12,14,41,48),(19,25,52,63),(20,26,53,64),(21,27,54,65),(22,28,49,66),(23,29,50,61),(24,30,51,62)]])
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