metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.213D6, (C2×D4).46D6, (C2×Q8).60D6, (C2×C12).270D4, C4.4D4.6S3, C6.104(C4○D8), C12.67(C4○D4), Q8⋊2Dic3⋊21C2, C12.6Q8⋊12C2, (C6×D4).62C22, (C6×Q8).54C22, C4.21(D4⋊2S3), (C4×C12).105C22, (C2×C12).374C23, D4⋊Dic3.13C2, C6.42(C4.4D4), C2.23(Q8.13D6), C2.9(C23.12D6), C4⋊Dic3.151C22, C3⋊4(C42.78C22), (C4×C3⋊C8)⋊11C2, (C2×C6).505(C2×D4), (C2×C3⋊C8).252C22, (C3×C4.4D4).4C2, (C2×C4).109(C3⋊D4), (C2×C4).474(C22×S3), C22.180(C2×C3⋊D4), SmallGroup(192,615)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.213D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 240 in 96 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42.78C22, C4×C3⋊C8, D4⋊Dic3, Q8⋊2Dic3, C12.6Q8, C3×C4.4D4, C42.213D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C4○D8, D4⋊2S3, C2×C3⋊D4, C42.78C22, C23.12D6, Q8.13D6, C42.213D6
►On 96 points
Generators in S96
(1 18 11 56)(2 16 12 60)(3 14 10 58)(4 17 9 55)(5 15 7 59)(6 13 8 57)(19 44 34 47)(20 65 35 62)(21 46 36 43)(22 61 31 64)(23 48 32 45)(24 63 33 66)(25 75 91 54)(26 80 92 68)(27 77 93 50)(28 82 94 70)(29 73 95 52)(30 84 96 72)(37 71 89 83)(38 53 90 74)(39 67 85 79)(40 49 86 76)(41 69 87 81)(42 51 88 78)
(1 21 7 24)(2 19 8 22)(3 23 9 20)(4 35 10 32)(5 33 11 36)(6 31 12 34)(13 64 60 47)(14 48 55 65)(15 66 56 43)(16 44 57 61)(17 62 58 45)(18 46 59 63)(25 42 39 28)(26 29 40 37)(27 38 41 30)(49 71 80 73)(50 74 81 72)(51 67 82 75)(52 76 83 68)(53 69 84 77)(54 78 79 70)(85 94 91 88)(86 89 92 95)(87 96 93 90)
(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 52 11 73)(2 54 12 75)(3 50 10 77)(4 69 9 81)(5 71 7 83)(6 67 8 79)(13 25 57 91)(14 87 58 41)(15 29 59 95)(16 85 60 39)(17 27 55 93)(18 89 56 37)(19 70 34 82)(20 74 35 53)(21 68 36 80)(22 78 31 51)(23 72 32 84)(24 76 33 49)(26 63 92 66)(28 61 94 64)(30 65 96 62)(38 48 90 45)(40 46 86 43)(42 44 88 47)
G:=sub<Sym(96)| (1,18,11,56)(2,16,12,60)(3,14,10,58)(4,17,9,55)(5,15,7,59)(6,13,8,57)(19,44,34,47)(20,65,35,62)(21,46,36,43)(22,61,31,64)(23,48,32,45)(24,63,33,66)(25,75,91,54)(26,80,92,68)(27,77,93,50)(28,82,94,70)(29,73,95,52)(30,84,96,72)(37,71,89,83)(38,53,90,74)(39,67,85,79)(40,49,86,76)(41,69,87,81)(42,51,88,78), (1,21,7,24)(2,19,8,22)(3,23,9,20)(4,35,10,32)(5,33,11,36)(6,31,12,34)(13,64,60,47)(14,48,55,65)(15,66,56,43)(16,44,57,61)(17,62,58,45)(18,46,59,63)(25,42,39,28)(26,29,40,37)(27,38,41,30)(49,71,80,73)(50,74,81,72)(51,67,82,75)(52,76,83,68)(53,69,84,77)(54,78,79,70)(85,94,91,88)(86,89,92,95)(87,96,93,90), (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), (1,52,11,73)(2,54,12,75)(3,50,10,77)(4,69,9,81)(5,71,7,83)(6,67,8,79)(13,25,57,91)(14,87,58,41)(15,29,59,95)(16,85,60,39)(17,27,55,93)(18,89,56,37)(19,70,34,82)(20,74,35,53)(21,68,36,80)(22,78,31,51)(23,72,32,84)(24,76,33,49)(26,63,92,66)(28,61,94,64)(30,65,96,62)(38,48,90,45)(40,46,86,43)(42,44,88,47)>;
G:=Group( (1,18,11,56)(2,16,12,60)(3,14,10,58)(4,17,9,55)(5,15,7,59)(6,13,8,57)(19,44,34,47)(20,65,35,62)(21,46,36,43)(22,61,31,64)(23,48,32,45)(24,63,33,66)(25,75,91,54)(26,80,92,68)(27,77,93,50)(28,82,94,70)(29,73,95,52)(30,84,96,72)(37,71,89,83)(38,53,90,74)(39,67,85,79)(40,49,86,76)(41,69,87,81)(42,51,88,78), (1,21,7,24)(2,19,8,22)(3,23,9,20)(4,35,10,32)(5,33,11,36)(6,31,12,34)(13,64,60,47)(14,48,55,65)(15,66,56,43)(16,44,57,61)(17,62,58,45)(18,46,59,63)(25,42,39,28)(26,29,40,37)(27,38,41,30)(49,71,80,73)(50,74,81,72)(51,67,82,75)(52,76,83,68)(53,69,84,77)(54,78,79,70)(85,94,91,88)(86,89,92,95)(87,96,93,90), (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), (1,52,11,73)(2,54,12,75)(3,50,10,77)(4,69,9,81)(5,71,7,83)(6,67,8,79)(13,25,57,91)(14,87,58,41)(15,29,59,95)(16,85,60,39)(17,27,55,93)(18,89,56,37)(19,70,34,82)(20,74,35,53)(21,68,36,80)(22,78,31,51)(23,72,32,84)(24,76,33,49)(26,63,92,66)(28,61,94,64)(30,65,96,62)(38,48,90,45)(40,46,86,43)(42,44,88,47) );
G=PermutationGroup([[(1,18,11,56),(2,16,12,60),(3,14,10,58),(4,17,9,55),(5,15,7,59),(6,13,8,57),(19,44,34,47),(20,65,35,62),(21,46,36,43),(22,61,31,64),(23,48,32,45),(24,63,33,66),(25,75,91,54),(26,80,92,68),(27,77,93,50),(28,82,94,70),(29,73,95,52),(30,84,96,72),(37,71,89,83),(38,53,90,74),(39,67,85,79),(40,49,86,76),(41,69,87,81),(42,51,88,78)], [(1,21,7,24),(2,19,8,22),(3,23,9,20),(4,35,10,32),(5,33,11,36),(6,31,12,34),(13,64,60,47),(14,48,55,65),(15,66,56,43),(16,44,57,61),(17,62,58,45),(18,46,59,63),(25,42,39,28),(26,29,40,37),(27,38,41,30),(49,71,80,73),(50,74,81,72),(51,67,82,75),(52,76,83,68),(53,69,84,77),(54,78,79,70),(85,94,91,88),(86,89,92,95),(87,96,93,90)], [(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)], [(1,52,11,73),(2,54,12,75),(3,50,10,77),(4,69,9,81),(5,71,7,83),(6,67,8,79),(13,25,57,91),(14,87,58,41),(15,29,59,95),(16,85,60,39),(17,27,55,93),(18,89,56,37),(19,70,34,82),(20,74,35,53),(21,68,36,80),(22,78,31,51),(23,72,32,84),(24,76,33,49),(26,63,92,66),(28,61,94,64),(30,65,96,62),(38,48,90,45),(40,46,86,43),(42,44,88,47)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | ··· | 2 | 8 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | D4⋊2S3 | Q8.13D6 |
| kernel | C42.213D6 | C4×C3⋊C8 | D4⋊Dic3 | Q8⋊2Dic3 | C12.6Q8 | C3×C4.4D4 | C4.4D4 | C2×C12 | C42 | C2×D4 | C2×Q8 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C42.213D6 ►in GL6(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 30 |
| 0 | 0 | 0 | 0 | 39 | 27 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 66 |
| 0 | 0 | 0 | 0 | 42 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 53 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 31 | 72 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 44 | 0 | 0 |
| 0 | 0 | 71 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 34 |
| 0 | 0 | 0 | 0 | 58 | 0 |
G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,39,0,0,0,0,30,27],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,42,0,0,0,0,66,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,64,53,0,0,0,0,0,65,0,0,0,0,0,0,1,31,0,0,0,0,0,72],[67,6,0,0,0,0,6,6,0,0,0,0,0,0,69,71,0,0,0,0,44,4,0,0,0,0,0,0,0,58,0,0,0,0,34,0] >;
C42.213D6 in GAP, Magma, Sage, TeX
C_4^2._{213}D_6 % in TeX
G:=Group("C4^2.213D6"); // GroupNames label
G:=SmallGroup(192,615);
// by ID
G=gap.SmallGroup(192,615);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,471,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations