metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.240D6, (C4×S3)⋊9D4, C4⋊Q8⋊19S3, D6.8(C2×D4), C4.38(S3×D4), C12⋊6(C4○D4), C4⋊C4.217D6, C12.70(C2×D4), C12⋊D4⋊39C2, C4⋊D12⋊17C2, (S3×C42)⋊14C2, C4⋊1(Q8⋊3S3), (C2×Q8).168D6, Dic3⋊5D4⋊43C2, C6.99(C22×D4), (C2×C6).269C24, D6⋊C4.50C22, Dic3.67(C2×D4), C12.23D4⋊26C2, (C2×C12).102C23, (C4×C12).210C22, (C6×Q8).136C22, (C2×D12).172C22, C22.290(S3×C23), C3⋊6(C22.26C24), (C22×S3).119C23, (C4×Dic3).258C22, (C2×Dic3).272C23, C2.72(C2×S3×D4), (C3×C4⋊Q8)⋊11C2, C6.120(C2×C4○D4), (C2×Q8⋊3S3)⋊12C2, (S3×C2×C4).143C22, C2.27(C2×Q8⋊3S3), (C3×C4⋊C4).212C22, (C2×C4).599(C22×S3), SmallGroup(192,1284)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.240D6
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c5 >
Subgroups: 848 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4×Dic3, C4×Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C22.26C24, S3×C42, C4⋊D12, Dic3⋊5D4, C12⋊D4, C12.23D4, C3×C4⋊Q8, C2×Q8⋊3S3, C42.240D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, S3×D4, Q8⋊3S3, S3×C23, C22.26C24, C2×S3×D4, C2×Q8⋊3S3, C42.240D6
►On 96 points
(1 28 52 93)(2 94 53 29)(3 30 54 95)(4 96 55 31)(5 32 56 85)(6 86 57 33)(7 34 58 87)(8 88 59 35)(9 36 60 89)(10 90 49 25)(11 26 50 91)(12 92 51 27)(13 75 65 39)(14 40 66 76)(15 77 67 41)(16 42 68 78)(17 79 69 43)(18 44 70 80)(19 81 71 45)(20 46 72 82)(21 83 61 47)(22 48 62 84)(23 73 63 37)(24 38 64 74)
(1 21 7 15)(2 16 8 22)(3 23 9 17)(4 18 10 24)(5 13 11 19)(6 20 12 14)(25 74 31 80)(26 81 32 75)(27 76 33 82)(28 83 34 77)(29 78 35 84)(30 73 36 79)(37 89 43 95)(38 96 44 90)(39 91 45 85)(40 86 46 92)(41 93 47 87)(42 88 48 94)(49 64 55 70)(50 71 56 65)(51 66 57 72)(52 61 58 67)(53 68 59 62)(54 63 60 69)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 19)(14 18)(15 17)(20 24)(21 23)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 96)(34 95)(35 94)(36 93)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 84)(49 51)(52 60)(53 59)(54 58)(55 57)(61 63)(64 72)(65 71)(66 70)(67 69)
G:=sub<Sym(96)| (1,28,52,93)(2,94,53,29)(3,30,54,95)(4,96,55,31)(5,32,56,85)(6,86,57,33)(7,34,58,87)(8,88,59,35)(9,36,60,89)(10,90,49,25)(11,26,50,91)(12,92,51,27)(13,75,65,39)(14,40,66,76)(15,77,67,41)(16,42,68,78)(17,79,69,43)(18,44,70,80)(19,81,71,45)(20,46,72,82)(21,83,61,47)(22,48,62,84)(23,73,63,37)(24,38,64,74), (1,21,7,15)(2,16,8,22)(3,23,9,17)(4,18,10,24)(5,13,11,19)(6,20,12,14)(25,74,31,80)(26,81,32,75)(27,76,33,82)(28,83,34,77)(29,78,35,84)(30,73,36,79)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,19)(14,18)(15,17)(20,24)(21,23)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,96)(34,95)(35,94)(36,93)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,84)(49,51)(52,60)(53,59)(54,58)(55,57)(61,63)(64,72)(65,71)(66,70)(67,69)>;
G:=Group( (1,28,52,93)(2,94,53,29)(3,30,54,95)(4,96,55,31)(5,32,56,85)(6,86,57,33)(7,34,58,87)(8,88,59,35)(9,36,60,89)(10,90,49,25)(11,26,50,91)(12,92,51,27)(13,75,65,39)(14,40,66,76)(15,77,67,41)(16,42,68,78)(17,79,69,43)(18,44,70,80)(19,81,71,45)(20,46,72,82)(21,83,61,47)(22,48,62,84)(23,73,63,37)(24,38,64,74), (1,21,7,15)(2,16,8,22)(3,23,9,17)(4,18,10,24)(5,13,11,19)(6,20,12,14)(25,74,31,80)(26,81,32,75)(27,76,33,82)(28,83,34,77)(29,78,35,84)(30,73,36,79)(37,89,43,95)(38,96,44,90)(39,91,45,85)(40,86,46,92)(41,93,47,87)(42,88,48,94)(49,64,55,70)(50,71,56,65)(51,66,57,72)(52,61,58,67)(53,68,59,62)(54,63,60,69), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,19)(14,18)(15,17)(20,24)(21,23)(25,92)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,96)(34,95)(35,94)(36,93)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,84)(49,51)(52,60)(53,59)(54,58)(55,57)(61,63)(64,72)(65,71)(66,70)(67,69) );
G=PermutationGroup([[(1,28,52,93),(2,94,53,29),(3,30,54,95),(4,96,55,31),(5,32,56,85),(6,86,57,33),(7,34,58,87),(8,88,59,35),(9,36,60,89),(10,90,49,25),(11,26,50,91),(12,92,51,27),(13,75,65,39),(14,40,66,76),(15,77,67,41),(16,42,68,78),(17,79,69,43),(18,44,70,80),(19,81,71,45),(20,46,72,82),(21,83,61,47),(22,48,62,84),(23,73,63,37),(24,38,64,74)], [(1,21,7,15),(2,16,8,22),(3,23,9,17),(4,18,10,24),(5,13,11,19),(6,20,12,14),(25,74,31,80),(26,81,32,75),(27,76,33,82),(28,83,34,77),(29,78,35,84),(30,73,36,79),(37,89,43,95),(38,96,44,90),(39,91,45,85),(40,86,46,92),(41,93,47,87),(42,88,48,94),(49,64,55,70),(50,71,56,65),(51,66,57,72),(52,61,58,67),(53,68,59,62),(54,63,60,69)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,19),(14,18),(15,17),(20,24),(21,23),(25,92),(26,91),(27,90),(28,89),(29,88),(30,87),(31,86),(32,85),(33,96),(34,95),(35,94),(36,93),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,84),(49,51),(52,60),(53,59),(54,58),(55,57),(61,63),(64,72),(65,71),(66,70),(67,69)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | S3×D4 | Q8⋊3S3 |
| kernel | C42.240D6 | S3×C42 | C4⋊D12 | Dic3⋊5D4 | C12⋊D4 | C12.23D4 | C3×C4⋊Q8 | C2×Q8⋊3S3 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 4 |
Matrix representation of C42.240D6 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 3 | 8 | 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 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [5,3,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,10,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
C42.240D6 in GAP, Magma, Sage, TeX
C_4^2._{240}D_6 % in TeX
G:=Group("C4^2.240D6"); // GroupNames label
G:=SmallGroup(192,1284);
// by ID
G=gap.SmallGroup(192,1284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^5>;
// generators/relations