metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.139D6, C6.882- 1+4, (C4×Dic6)⋊44C2, (Q8×Dic3)⋊18C2, (C2×D4).169D6, (C2×Q8).159D6, C22⋊C4.33D6, C4.4D4.8S3, (C2×C6).215C24, (C2×C12).77C23, C2.49(Q8○D12), (D4×Dic3).14C2, Dic3⋊Q8⋊22C2, C12.124(C4○D4), C4.15(D4⋊2S3), (C4×C12).184C22, C23.12D6.9C2, (C6×D4).151C22, C23.8D6⋊37C2, C23.47(C22×S3), (C22×C6).45C23, (C6×Q8).124C22, Dic3.28(C4○D4), Dic3.D4⋊38C2, C23.16D6⋊18C2, Dic3⋊C4.48C22, C4⋊Dic3.233C22, C22.236(S3×C23), C3⋊6(C22.50C24), (C2×Dic3).252C23, (C2×Dic6).296C22, (C4×Dic3).131C22, C6.D4.52C22, (C22×Dic3).140C22, C6.93(C2×C4○D4), C2.74(S3×C4○D4), C2.55(C2×D4⋊2S3), (C3×C4.4D4).6C2, (C2×C4).299(C22×S3), (C3×C22⋊C4).62C22, SmallGroup(192,1230)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.139D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >
Subgroups: 448 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C22×Dic3, C6×D4, C6×Q8, C22.50C24, C4×Dic6, C23.16D6, Dic3.D4, C23.8D6, D4×Dic3, C23.12D6, Dic3⋊Q8, Q8×Dic3, C3×C4.4D4, C42.139D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, D4⋊2S3, S3×C23, C22.50C24, C2×D4⋊2S3, S3×C4○D4, Q8○D12, C42.139D6
►On 96 points
(1 43 24 46)(2 47 22 44)(3 45 23 48)(4 38 7 41)(5 42 8 39)(6 40 9 37)(10 69 13 72)(11 67 14 70)(12 71 15 68)(16 63 20 66)(17 61 21 64)(18 65 19 62)(25 60 52 34)(26 35 53 55)(27 56 54 36)(28 31 49 57)(29 58 50 32)(30 33 51 59)(73 89 82 95)(74 96 83 90)(75 85 84 91)(76 92 79 86)(77 87 80 93)(78 94 81 88)
(1 49 7 25)(2 53 8 29)(3 51 9 27)(4 52 24 28)(5 50 22 26)(6 54 23 30)(10 84 18 78)(11 82 16 76)(12 80 17 74)(13 75 19 81)(14 73 20 79)(15 77 21 83)(31 38 34 46)(32 44 35 42)(33 40 36 48)(37 56 45 59)(39 58 47 55)(41 60 43 57)(61 96 71 93)(62 88 72 85)(63 92 67 95)(64 90 68 87)(65 94 69 91)(66 86 70 89)
(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 20 4 11)(2 19 5 10)(3 21 6 12)(7 14 24 16)(8 13 22 18)(9 15 23 17)(25 73 28 76)(26 78 29 75)(27 77 30 74)(31 86 60 95)(32 85 55 94)(33 90 56 93)(34 89 57 92)(35 88 58 91)(36 87 59 96)(37 71 48 64)(38 70 43 63)(39 69 44 62)(40 68 45 61)(41 67 46 66)(42 72 47 65)(49 79 52 82)(50 84 53 81)(51 83 54 80)
G:=sub<Sym(96)| (1,43,24,46)(2,47,22,44)(3,45,23,48)(4,38,7,41)(5,42,8,39)(6,40,9,37)(10,69,13,72)(11,67,14,70)(12,71,15,68)(16,63,20,66)(17,61,21,64)(18,65,19,62)(25,60,52,34)(26,35,53,55)(27,56,54,36)(28,31,49,57)(29,58,50,32)(30,33,51,59)(73,89,82,95)(74,96,83,90)(75,85,84,91)(76,92,79,86)(77,87,80,93)(78,94,81,88), (1,49,7,25)(2,53,8,29)(3,51,9,27)(4,52,24,28)(5,50,22,26)(6,54,23,30)(10,84,18,78)(11,82,16,76)(12,80,17,74)(13,75,19,81)(14,73,20,79)(15,77,21,83)(31,38,34,46)(32,44,35,42)(33,40,36,48)(37,56,45,59)(39,58,47,55)(41,60,43,57)(61,96,71,93)(62,88,72,85)(63,92,67,95)(64,90,68,87)(65,94,69,91)(66,86,70,89), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,20,4,11)(2,19,5,10)(3,21,6,12)(7,14,24,16)(8,13,22,18)(9,15,23,17)(25,73,28,76)(26,78,29,75)(27,77,30,74)(31,86,60,95)(32,85,55,94)(33,90,56,93)(34,89,57,92)(35,88,58,91)(36,87,59,96)(37,71,48,64)(38,70,43,63)(39,69,44,62)(40,68,45,61)(41,67,46,66)(42,72,47,65)(49,79,52,82)(50,84,53,81)(51,83,54,80)>;
G:=Group( (1,43,24,46)(2,47,22,44)(3,45,23,48)(4,38,7,41)(5,42,8,39)(6,40,9,37)(10,69,13,72)(11,67,14,70)(12,71,15,68)(16,63,20,66)(17,61,21,64)(18,65,19,62)(25,60,52,34)(26,35,53,55)(27,56,54,36)(28,31,49,57)(29,58,50,32)(30,33,51,59)(73,89,82,95)(74,96,83,90)(75,85,84,91)(76,92,79,86)(77,87,80,93)(78,94,81,88), (1,49,7,25)(2,53,8,29)(3,51,9,27)(4,52,24,28)(5,50,22,26)(6,54,23,30)(10,84,18,78)(11,82,16,76)(12,80,17,74)(13,75,19,81)(14,73,20,79)(15,77,21,83)(31,38,34,46)(32,44,35,42)(33,40,36,48)(37,56,45,59)(39,58,47,55)(41,60,43,57)(61,96,71,93)(62,88,72,85)(63,92,67,95)(64,90,68,87)(65,94,69,91)(66,86,70,89), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,20,4,11)(2,19,5,10)(3,21,6,12)(7,14,24,16)(8,13,22,18)(9,15,23,17)(25,73,28,76)(26,78,29,75)(27,77,30,74)(31,86,60,95)(32,85,55,94)(33,90,56,93)(34,89,57,92)(35,88,58,91)(36,87,59,96)(37,71,48,64)(38,70,43,63)(39,69,44,62)(40,68,45,61)(41,67,46,66)(42,72,47,65)(49,79,52,82)(50,84,53,81)(51,83,54,80) );
G=PermutationGroup([[(1,43,24,46),(2,47,22,44),(3,45,23,48),(4,38,7,41),(5,42,8,39),(6,40,9,37),(10,69,13,72),(11,67,14,70),(12,71,15,68),(16,63,20,66),(17,61,21,64),(18,65,19,62),(25,60,52,34),(26,35,53,55),(27,56,54,36),(28,31,49,57),(29,58,50,32),(30,33,51,59),(73,89,82,95),(74,96,83,90),(75,85,84,91),(76,92,79,86),(77,87,80,93),(78,94,81,88)], [(1,49,7,25),(2,53,8,29),(3,51,9,27),(4,52,24,28),(5,50,22,26),(6,54,23,30),(10,84,18,78),(11,82,16,76),(12,80,17,74),(13,75,19,81),(14,73,20,79),(15,77,21,83),(31,38,34,46),(32,44,35,42),(33,40,36,48),(37,56,45,59),(39,58,47,55),(41,60,43,57),(61,96,71,93),(62,88,72,85),(63,92,67,95),(64,90,68,87),(65,94,69,91),(66,86,70,89)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,20,4,11),(2,19,5,10),(3,21,6,12),(7,14,24,16),(8,13,22,18),(9,15,23,17),(25,73,28,76),(26,78,29,75),(27,77,30,74),(31,86,60,95),(32,85,55,94),(33,90,56,93),(34,89,57,92),(35,88,58,91),(36,87,59,96),(37,71,48,64),(38,70,43,63),(39,69,44,62),(40,68,45,61),(41,67,46,66),(42,72,47,65),(49,79,52,82),(50,84,53,81),(51,83,54,80)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2- 1+4 | D4⋊2S3 | S3×C4○D4 | Q8○D12 |
| kernel | C42.139D6 | C4×Dic6 | C23.16D6 | Dic3.D4 | C23.8D6 | D4×Dic3 | C23.12D6 | Dic3⋊Q8 | Q8×Dic3 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C12 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C42.139D6 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 7 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,11,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,9,11,0,0,0,0,0,3,0,0,0,0,0,0,1,5,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,11,7,0,0,0,0,7,2,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
C42.139D6 in GAP, Magma, Sage, TeX
C_4^2._{139}D_6 % in TeX
G:=Group("C4^2.139D6"); // GroupNames label
G:=SmallGroup(192,1230);
// by ID
G=gap.SmallGroup(192,1230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,387,100,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations