metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.30D6, C4⋊C8⋊15S3, (C4×S3).9Q8, C4.56(S3×Q8), D6⋊C4.14C4, D6.2(C4⋊C4), C12⋊C8⋊15C2, (C4×S3).49D4, C4.209(S3×D4), (C2×C8).184D6, Dic3⋊C8⋊29C2, C6.13(C8○D4), C12.368(C2×D4), C12.114(C2×Q8), C2.15(C8○D12), Dic3⋊C4.14C4, Dic3.3(C4⋊C4), C42⋊2S3.1C2, (C4×C12).65C22, C2.14(D12.C4), (C2×C12).836C23, (C2×C24).258C22, C3⋊1(C42.6C22), (C4×Dic3).185C22, C6.9(C2×C4⋊C4), (C3×C4⋊C8)⋊25C2, C2.10(S3×C4⋊C4), (S3×C2×C8).14C2, (C2×C4).36(C4×S3), (C2×C8⋊S3).9C2, (C2×C12).44(C2×C4), C22.114(S3×C2×C4), (C2×C3⋊C8).308C22, (S3×C2×C4).281C22, (C2×C6).91(C22×C4), (C22×S3).39(C2×C4), (C2×C4).778(C22×S3), (C2×Dic3).54(C2×C4), SmallGroup(192,398)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.30D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b2c5 >
Subgroups: 248 in 114 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C4⋊C8, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C42.6C22, C12⋊C8, Dic3⋊C8, C3×C4⋊C8, C42⋊2S3, S3×C2×C8, C2×C8⋊S3, C42.30D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, C8○D4, S3×C2×C4, S3×D4, S3×Q8, C42.6C22, S3×C4⋊C4, C8○D12, D12.C4, C42.30D6
►On 96 points
(1 88 72 27)(2 28 49 89)(3 90 50 29)(4 30 51 91)(5 92 52 31)(6 32 53 93)(7 94 54 33)(8 34 55 95)(9 96 56 35)(10 36 57 73)(11 74 58 37)(12 38 59 75)(13 76 60 39)(14 40 61 77)(15 78 62 41)(16 42 63 79)(17 80 64 43)(18 44 65 81)(19 82 66 45)(20 46 67 83)(21 84 68 47)(22 48 69 85)(23 86 70 25)(24 26 71 87)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)
(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 18 19 12 13 6 7 24)(2 11 20 5 14 23 8 17)(3 4 21 22 15 16 9 10)(25 46 43 40 37 34 31 28)(26 39 44 33 38 27 32 45)(29 42 47 36 41 30 35 48)(49 58 67 52 61 70 55 64)(50 51 68 69 62 63 56 57)(53 54 71 72 65 66 59 60)(73 78 91 96 85 90 79 84)(74 95 92 89 86 83 80 77)(75 88 93 82 87 76 81 94)
G:=sub<Sym(96)| (1,88,72,27)(2,28,49,89)(3,90,50,29)(4,30,51,91)(5,92,52,31)(6,32,53,93)(7,94,54,33)(8,34,55,95)(9,96,56,35)(10,36,57,73)(11,74,58,37)(12,38,59,75)(13,76,60,39)(14,40,61,77)(15,78,62,41)(16,42,63,79)(17,80,64,43)(18,44,65,81)(19,82,66,45)(20,46,67,83)(21,84,68,47)(22,48,69,85)(23,86,70,25)(24,26,71,87), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,46,43,40,37,34,31,28)(26,39,44,33,38,27,32,45)(29,42,47,36,41,30,35,48)(49,58,67,52,61,70,55,64)(50,51,68,69,62,63,56,57)(53,54,71,72,65,66,59,60)(73,78,91,96,85,90,79,84)(74,95,92,89,86,83,80,77)(75,88,93,82,87,76,81,94)>;
G:=Group( (1,88,72,27)(2,28,49,89)(3,90,50,29)(4,30,51,91)(5,92,52,31)(6,32,53,93)(7,94,54,33)(8,34,55,95)(9,96,56,35)(10,36,57,73)(11,74,58,37)(12,38,59,75)(13,76,60,39)(14,40,61,77)(15,78,62,41)(16,42,63,79)(17,80,64,43)(18,44,65,81)(19,82,66,45)(20,46,67,83)(21,84,68,47)(22,48,69,85)(23,86,70,25)(24,26,71,87), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,46,43,40,37,34,31,28)(26,39,44,33,38,27,32,45)(29,42,47,36,41,30,35,48)(49,58,67,52,61,70,55,64)(50,51,68,69,62,63,56,57)(53,54,71,72,65,66,59,60)(73,78,91,96,85,90,79,84)(74,95,92,89,86,83,80,77)(75,88,93,82,87,76,81,94) );
G=PermutationGroup([[(1,88,72,27),(2,28,49,89),(3,90,50,29),(4,30,51,91),(5,92,52,31),(6,32,53,93),(7,94,54,33),(8,34,55,95),(9,96,56,35),(10,36,57,73),(11,74,58,37),(12,38,59,75),(13,76,60,39),(14,40,61,77),(15,78,62,41),(16,42,63,79),(17,80,64,43),(18,44,65,81),(19,82,66,45),(20,46,67,83),(21,84,68,47),(22,48,69,85),(23,86,70,25),(24,26,71,87)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84)], [(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)], [(1,18,19,12,13,6,7,24),(2,11,20,5,14,23,8,17),(3,4,21,22,15,16,9,10),(25,46,43,40,37,34,31,28),(26,39,44,33,38,27,32,45),(29,42,47,36,41,30,35,48),(49,58,67,52,61,70,55,64),(50,51,68,69,62,63,56,57),(53,54,71,72,65,66,59,60),(73,78,91,96,85,90,79,84),(74,95,92,89,86,83,80,77),(75,88,93,82,87,76,81,94)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C4 | C4 | S3 | D4 | Q8 | D6 | D6 | C4×S3 | C8○D4 | C8○D12 | S3×D4 | S3×Q8 | D12.C4 |
| kernel | C42.30D6 | C12⋊C8 | Dic3⋊C8 | C3×C4⋊C8 | C42⋊2S3 | S3×C2×C8 | C2×C8⋊S3 | Dic3⋊C4 | D6⋊C4 | C4⋊C8 | C4×S3 | C4×S3 | C42 | C2×C8 | C2×C4 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 1 | 2 | 4 | 8 | 8 | 1 | 1 | 2 |
Matrix representation of C42.30D6 ►in GL4(𝔽73) generated by
| 1 | 3 | 0 | 0 |
| 48 | 72 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 13 | 43 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 51 | 0 | 0 | 0 |
| 39 | 22 | 0 | 0 |
| 0 | 0 | 22 | 22 |
| 0 | 0 | 51 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 22 | 22 |
| 0 | 0 | 0 | 51 |
G:=sub<GL(4,GF(73))| [1,48,0,0,3,72,0,0,0,0,30,13,0,0,60,43],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[51,39,0,0,0,22,0,0,0,0,22,51,0,0,22,0],[51,0,0,0,0,51,0,0,0,0,22,0,0,0,22,51] >;
C42.30D6 in GAP, Magma, Sage, TeX
C_4^2._{30}D_6 % in TeX
G:=Group("C4^2.30D6"); // GroupNames label
G:=SmallGroup(192,398);
// by ID
G=gap.SmallGroup(192,398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^5>;
// generators/relations