metabelian, supersoluble, monomial
Aliases: C122⋊7C2, C12.58D12, C62.109D4, (C4×C12)⋊8S3, C32⋊9C4≀C2, C12⋊S3⋊4C4, C12.60(C4×S3), C42⋊4(C3⋊S3), C32⋊4Q8⋊4C4, C6.20(D6⋊C4), (C3×C12).138D4, (C2×C12).396D6, C3⋊2(C42⋊4S3), C12.58D6⋊1C2, C4.17(C12⋊S3), (C6×C12).306C22, C12.59D6.1C2, C2.3(C6.11D12), C22.7(C32⋊7D4), C4.6(C4×C3⋊S3), (C3×C12).91(C2×C4), (C2×C6).85(C3⋊D4), (C3×C6).51(C22⋊C4), (C2×C4).70(C2×C3⋊S3), SmallGroup(288,280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C122⋊C2
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b9, cbc=b-1 >
Subgroups: 508 in 132 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C4≀C2, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C4.Dic3, C4×C12, C4○D12, C32⋊4C8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C6×C12, C42⋊4S3, C12.58D6, C122, C12.59D6, C122⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C4≀C2, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C42⋊4S3, C6.11D12, C122⋊C2
►On 72 points
(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 25 12 35 15 24 4 28 9 32 18 21)(2 26 7 36 16 19 5 29 10 33 13 22)(3 27 8 31 17 20 6 30 11 34 14 23)(37 61 49 40 64 52 43 67 55 46 70 58)(38 62 50 41 65 53 44 68 56 47 71 59)(39 63 51 42 66 54 45 69 57 48 72 60)
(1 65)(2 67)(3 69)(4 71)(5 61)(6 63)(7 52)(8 54)(9 56)(10 58)(11 60)(12 50)(13 46)(14 48)(15 38)(16 40)(17 42)(18 44)(19 49)(20 51)(21 53)(22 55)(23 57)(24 59)(25 41)(26 43)(27 45)(28 47)(29 37)(30 39)(31 66)(32 68)(33 70)(34 72)(35 62)(36 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), (1,25,12,35,15,24,4,28,9,32,18,21)(2,26,7,36,16,19,5,29,10,33,13,22)(3,27,8,31,17,20,6,30,11,34,14,23)(37,61,49,40,64,52,43,67,55,46,70,58)(38,62,50,41,65,53,44,68,56,47,71,59)(39,63,51,42,66,54,45,69,57,48,72,60), (1,65)(2,67)(3,69)(4,71)(5,61)(6,63)(7,52)(8,54)(9,56)(10,58)(11,60)(12,50)(13,46)(14,48)(15,38)(16,40)(17,42)(18,44)(19,49)(20,51)(21,53)(22,55)(23,57)(24,59)(25,41)(26,43)(27,45)(28,47)(29,37)(30,39)(31,66)(32,68)(33,70)(34,72)(35,62)(36,64)>;
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), (1,25,12,35,15,24,4,28,9,32,18,21)(2,26,7,36,16,19,5,29,10,33,13,22)(3,27,8,31,17,20,6,30,11,34,14,23)(37,61,49,40,64,52,43,67,55,46,70,58)(38,62,50,41,65,53,44,68,56,47,71,59)(39,63,51,42,66,54,45,69,57,48,72,60), (1,65)(2,67)(3,69)(4,71)(5,61)(6,63)(7,52)(8,54)(9,56)(10,58)(11,60)(12,50)(13,46)(14,48)(15,38)(16,40)(17,42)(18,44)(19,49)(20,51)(21,53)(22,55)(23,57)(24,59)(25,41)(26,43)(27,45)(28,47)(29,37)(30,39)(31,66)(32,68)(33,70)(34,72)(35,62)(36,64) );
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)], [(1,25,12,35,15,24,4,28,9,32,18,21),(2,26,7,36,16,19,5,29,10,33,13,22),(3,27,8,31,17,20,6,30,11,34,14,23),(37,61,49,40,64,52,43,67,55,46,70,58),(38,62,50,41,65,53,44,68,56,47,71,59),(39,63,51,42,66,54,45,69,57,48,72,60)], [(1,65),(2,67),(3,69),(4,71),(5,61),(6,63),(7,52),(8,54),(9,56),(10,58),(11,60),(12,50),(13,46),(14,48),(15,38),(16,40),(17,42),(18,44),(19,49),(20,51),(21,53),(22,55),(23,57),(24,59),(25,41),(26,43),(27,45),(28,47),(29,37),(30,39),(31,66),(32,68),(33,70),(34,72),(35,62),(36,64)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6L | 8A | 8B | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 36 | 2 | ··· | 2 | 36 | 36 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C42⋊4S3 |
| kernel | C122⋊C2 | C12.58D6 | C122 | C12.59D6 | C32⋊4Q8 | C12⋊S3 | C4×C12 | C3×C12 | C62 | C2×C12 | C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 4 | 32 |
Matrix representation of C122⋊C2 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [9,0,0,0,0,70,0,0,0,0,1,0,0,0,0,72],[46,0,0,0,0,27,0,0,0,0,65,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C122⋊C2 in GAP, Magma, Sage, TeX
C_{12}^2\rtimes C_2 % in TeX
G:=Group("C12^2:C2"); // GroupNames label
G:=SmallGroup(288,280);
// by ID
G=gap.SmallGroup(288,280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5*b^9,c*b*c=b^-1>;
// generators/relations