metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.95D6, C6.972+ 1+4, Dic3⋊D4⋊3C2, (C4×D12)⋊10C2, C4⋊C4.272D6, C12⋊D4⋊12C2, C12⋊7D4⋊29C2, C42⋊3S3⋊4C2, C23.9D6⋊4C2, C2.9(D4○D12), (C2×C6).74C24, D6⋊C4.3C22, C22⋊C4.98D6, Dic3⋊5D4⋊12C2, C42⋊C2⋊14S3, D6.28(C4○D4), C4.96(C4○D12), (C4×C12).25C22, C4.Dic6⋊13C2, (C22×C4).211D6, C12.198(C4○D4), (C2×C12).149C23, C23.96(C22×S3), (C2×D12).137C22, Dic3⋊C4.98C22, (C22×S3).22C23, C4⋊Dic3.195C22, C22.103(S3×C23), (C22×C6).144C23, (C4×Dic3).70C22, (C2×Dic3).27C23, (C22×C12).232C22, C3⋊2(C22.47C24), C6.D4.97C22, (S3×C4⋊C4)⋊13C2, (C4×C3⋊D4)⋊12C2, C2.13(S3×C4○D4), C2.33(C2×C4○D12), C6.134(C2×C4○D4), (S3×C2×C4).61C22, (C3×C42⋊C2)⋊16C2, (C3×C4⋊C4).310C22, (C2×C4).276(C22×S3), (C2×C3⋊D4).104C22, (C3×C22⋊C4).114C22, SmallGroup(192,1089)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.95D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 632 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22.47C24, C4×D12, C42⋊3S3, C23.9D6, Dic3⋊D4, C4.Dic6, S3×C4⋊C4, Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C12⋊7D4, C3×C42⋊C2, C42.95D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.47C24, C2×C4○D12, S3×C4○D4, D4○D12, C42.95D6
►On 96 points
(1 46 80 94)(2 41 81 89)(3 48 82 96)(4 43 83 91)(5 38 84 86)(6 45 73 93)(7 40 74 88)(8 47 75 95)(9 42 76 90)(10 37 77 85)(11 44 78 92)(12 39 79 87)(13 65 29 52)(14 72 30 59)(15 67 31 54)(16 62 32 49)(17 69 33 56)(18 64 34 51)(19 71 35 58)(20 66 36 53)(21 61 25 60)(22 68 26 55)(23 63 27 50)(24 70 28 57)
(1 17 7 23)(2 18 8 24)(3 19 9 13)(4 20 10 14)(5 21 11 15)(6 22 12 16)(25 78 31 84)(26 79 32 73)(27 80 33 74)(28 81 34 75)(29 82 35 76)(30 83 36 77)(37 72 43 66)(38 61 44 67)(39 62 45 68)(40 63 46 69)(41 64 47 70)(42 65 48 71)(49 93 55 87)(50 94 56 88)(51 95 57 89)(52 96 58 90)(53 85 59 91)(54 86 60 92)
(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 55 7 49)(2 60 8 54)(3 53 9 59)(4 58 10 52)(5 51 11 57)(6 56 12 50)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)(25 89 31 95)(26 94 32 88)(27 87 33 93)(28 92 34 86)(29 85 35 91)(30 90 36 96)(61 75 67 81)(62 80 68 74)(63 73 69 79)(64 78 70 84)(65 83 71 77)(66 76 72 82)
G:=sub<Sym(96)| (1,46,80,94)(2,41,81,89)(3,48,82,96)(4,43,83,91)(5,38,84,86)(6,45,73,93)(7,40,74,88)(8,47,75,95)(9,42,76,90)(10,37,77,85)(11,44,78,92)(12,39,79,87)(13,65,29,52)(14,72,30,59)(15,67,31,54)(16,62,32,49)(17,69,33,56)(18,64,34,51)(19,71,35,58)(20,66,36,53)(21,61,25,60)(22,68,26,55)(23,63,27,50)(24,70,28,57), (1,17,7,23)(2,18,8,24)(3,19,9,13)(4,20,10,14)(5,21,11,15)(6,22,12,16)(25,78,31,84)(26,79,32,73)(27,80,33,74)(28,81,34,75)(29,82,35,76)(30,83,36,77)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,93,55,87)(50,94,56,88)(51,95,57,89)(52,96,58,90)(53,85,59,91)(54,86,60,92), (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), (1,55,7,49)(2,60,8,54)(3,53,9,59)(4,58,10,52)(5,51,11,57)(6,56,12,50)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)(25,89,31,95)(26,94,32,88)(27,87,33,93)(28,92,34,86)(29,85,35,91)(30,90,36,96)(61,75,67,81)(62,80,68,74)(63,73,69,79)(64,78,70,84)(65,83,71,77)(66,76,72,82)>;
G:=Group( (1,46,80,94)(2,41,81,89)(3,48,82,96)(4,43,83,91)(5,38,84,86)(6,45,73,93)(7,40,74,88)(8,47,75,95)(9,42,76,90)(10,37,77,85)(11,44,78,92)(12,39,79,87)(13,65,29,52)(14,72,30,59)(15,67,31,54)(16,62,32,49)(17,69,33,56)(18,64,34,51)(19,71,35,58)(20,66,36,53)(21,61,25,60)(22,68,26,55)(23,63,27,50)(24,70,28,57), (1,17,7,23)(2,18,8,24)(3,19,9,13)(4,20,10,14)(5,21,11,15)(6,22,12,16)(25,78,31,84)(26,79,32,73)(27,80,33,74)(28,81,34,75)(29,82,35,76)(30,83,36,77)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,93,55,87)(50,94,56,88)(51,95,57,89)(52,96,58,90)(53,85,59,91)(54,86,60,92), (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), (1,55,7,49)(2,60,8,54)(3,53,9,59)(4,58,10,52)(5,51,11,57)(6,56,12,50)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)(25,89,31,95)(26,94,32,88)(27,87,33,93)(28,92,34,86)(29,85,35,91)(30,90,36,96)(61,75,67,81)(62,80,68,74)(63,73,69,79)(64,78,70,84)(65,83,71,77)(66,76,72,82) );
G=PermutationGroup([[(1,46,80,94),(2,41,81,89),(3,48,82,96),(4,43,83,91),(5,38,84,86),(6,45,73,93),(7,40,74,88),(8,47,75,95),(9,42,76,90),(10,37,77,85),(11,44,78,92),(12,39,79,87),(13,65,29,52),(14,72,30,59),(15,67,31,54),(16,62,32,49),(17,69,33,56),(18,64,34,51),(19,71,35,58),(20,66,36,53),(21,61,25,60),(22,68,26,55),(23,63,27,50),(24,70,28,57)], [(1,17,7,23),(2,18,8,24),(3,19,9,13),(4,20,10,14),(5,21,11,15),(6,22,12,16),(25,78,31,84),(26,79,32,73),(27,80,33,74),(28,81,34,75),(29,82,35,76),(30,83,36,77),(37,72,43,66),(38,61,44,67),(39,62,45,68),(40,63,46,69),(41,64,47,70),(42,65,48,71),(49,93,55,87),(50,94,56,88),(51,95,57,89),(52,96,58,90),(53,85,59,91),(54,86,60,92)], [(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)], [(1,55,7,49),(2,60,8,54),(3,53,9,59),(4,58,10,52),(5,51,11,57),(6,56,12,50),(13,37,19,43),(14,42,20,48),(15,47,21,41),(16,40,22,46),(17,45,23,39),(18,38,24,44),(25,89,31,95),(26,94,32,88),(27,87,33,93),(28,92,34,86),(29,85,35,91),(30,90,36,96),(61,75,67,81),(62,80,68,74),(63,73,69,79),(64,78,70,84),(65,83,71,77),(66,76,72,82)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ 1+4 | S3×C4○D4 | D4○D12 |
| kernel | C42.95D6 | C4×D12 | C42⋊3S3 | C23.9D6 | Dic3⋊D4 | C4.Dic6 | S3×C4⋊C4 | Dic3⋊5D4 | C12⋊D4 | C4×C3⋊D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.95D6 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 4 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 3 | 10 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,8],[2,9,0,0,4,11,0,0,0,0,8,0,0,0,0,8],[12,1,0,0,12,0,0,0,0,0,0,12,0,0,1,0],[3,7,0,0,10,10,0,0,0,0,0,12,0,0,1,0] >;
C42.95D6 in GAP, Magma, Sage, TeX
C_4^2._{95}D_6 % in TeX
G:=Group("C4^2.95D6"); // GroupNames label
G:=SmallGroup(192,1089);
// by ID
G=gap.SmallGroup(192,1089);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations