metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.106D6, C6.572- 1+4, C4⋊C4.314D6, (C4×D4).13S3, C12⋊2Q8⋊22C2, (C4×Dic6)⋊28C2, (C2×D4).210D6, (D4×C12).14C2, (C2×C6).86C24, C4.15(C4○D12), C22⋊C4.107D6, C2.15(Q8○D12), (C22×C4).221D6, C23.8D6⋊7C2, Dic6⋊C4⋊14C2, C12.109(C4○D4), C12.48D4⋊19C2, (C2×C12).155C23, (C4×C12).148C22, C23.12D6.7C2, (C6×D4).250C22, C23.26D6⋊7C2, C4.116(D4⋊2S3), C4⋊Dic3.297C22, C22.114(S3×C23), (C22×C6).156C23, C23.102(C22×S3), (C2×Dic3).36C23, (C4×Dic3).73C22, Dic3⋊C4.109C22, (C22×C12).105C22, C3⋊2(C22.50C24), (C2×Dic6).237C22, C6.D4.103C22, C6.38(C2×C4○D4), C2.42(C2×C4○D12), C2.20(C2×D4⋊2S3), (C3×C4⋊C4).322C22, (C2×C4).281(C22×S3), (C3×C22⋊C4).120C22, SmallGroup(192,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.106D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 440 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×C12, C6×D4, C22.50C24, C4×Dic6, C12⋊2Q8, C23.8D6, Dic6⋊C4, C12.48D4, C23.26D6, C23.12D6, D4×C12, C42.106D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, D4⋊2S3, S3×C23, C22.50C24, C2×C4○D12, C2×D4⋊2S3, Q8○D12, C42.106D6
►On 96 points
(1 43 74 94)(2 95 75 44)(3 45 76 96)(4 85 77 46)(5 47 78 86)(6 87 79 48)(7 37 80 88)(8 89 81 38)(9 39 82 90)(10 91 83 40)(11 41 84 92)(12 93 73 42)(13 63 51 35)(14 36 52 64)(15 65 53 25)(16 26 54 66)(17 67 55 27)(18 28 56 68)(19 69 57 29)(20 30 58 70)(21 71 59 31)(22 32 60 72)(23 61 49 33)(24 34 50 62)
(1 46 7 40)(2 47 8 41)(3 48 9 42)(4 37 10 43)(5 38 11 44)(6 39 12 45)(13 72 19 66)(14 61 20 67)(15 62 21 68)(16 63 22 69)(17 64 23 70)(18 65 24 71)(25 50 31 56)(26 51 32 57)(27 52 33 58)(28 53 34 59)(29 54 35 60)(30 55 36 49)(73 96 79 90)(74 85 80 91)(75 86 81 92)(76 87 82 93)(77 88 83 94)(78 89 84 95)
(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 74 58)(2 13 75 51)(3 18 76 56)(4 23 77 49)(5 16 78 54)(6 21 79 59)(7 14 80 52)(8 19 81 57)(9 24 82 50)(10 17 83 55)(11 22 84 60)(12 15 73 53)(25 42 65 93)(26 47 66 86)(27 40 67 91)(28 45 68 96)(29 38 69 89)(30 43 70 94)(31 48 71 87)(32 41 72 92)(33 46 61 85)(34 39 62 90)(35 44 63 95)(36 37 64 88)
G:=sub<Sym(96)| (1,43,74,94)(2,95,75,44)(3,45,76,96)(4,85,77,46)(5,47,78,86)(6,87,79,48)(7,37,80,88)(8,89,81,38)(9,39,82,90)(10,91,83,40)(11,41,84,92)(12,93,73,42)(13,63,51,35)(14,36,52,64)(15,65,53,25)(16,26,54,66)(17,67,55,27)(18,28,56,68)(19,69,57,29)(20,30,58,70)(21,71,59,31)(22,32,60,72)(23,61,49,33)(24,34,50,62), (1,46,7,40)(2,47,8,41)(3,48,9,42)(4,37,10,43)(5,38,11,44)(6,39,12,45)(13,72,19,66)(14,61,20,67)(15,62,21,68)(16,63,22,69)(17,64,23,70)(18,65,24,71)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(73,96,79,90)(74,85,80,91)(75,86,81,92)(76,87,82,93)(77,88,83,94)(78,89,84,95), (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), (1,20,74,58)(2,13,75,51)(3,18,76,56)(4,23,77,49)(5,16,78,54)(6,21,79,59)(7,14,80,52)(8,19,81,57)(9,24,82,50)(10,17,83,55)(11,22,84,60)(12,15,73,53)(25,42,65,93)(26,47,66,86)(27,40,67,91)(28,45,68,96)(29,38,69,89)(30,43,70,94)(31,48,71,87)(32,41,72,92)(33,46,61,85)(34,39,62,90)(35,44,63,95)(36,37,64,88)>;
G:=Group( (1,43,74,94)(2,95,75,44)(3,45,76,96)(4,85,77,46)(5,47,78,86)(6,87,79,48)(7,37,80,88)(8,89,81,38)(9,39,82,90)(10,91,83,40)(11,41,84,92)(12,93,73,42)(13,63,51,35)(14,36,52,64)(15,65,53,25)(16,26,54,66)(17,67,55,27)(18,28,56,68)(19,69,57,29)(20,30,58,70)(21,71,59,31)(22,32,60,72)(23,61,49,33)(24,34,50,62), (1,46,7,40)(2,47,8,41)(3,48,9,42)(4,37,10,43)(5,38,11,44)(6,39,12,45)(13,72,19,66)(14,61,20,67)(15,62,21,68)(16,63,22,69)(17,64,23,70)(18,65,24,71)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(73,96,79,90)(74,85,80,91)(75,86,81,92)(76,87,82,93)(77,88,83,94)(78,89,84,95), (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), (1,20,74,58)(2,13,75,51)(3,18,76,56)(4,23,77,49)(5,16,78,54)(6,21,79,59)(7,14,80,52)(8,19,81,57)(9,24,82,50)(10,17,83,55)(11,22,84,60)(12,15,73,53)(25,42,65,93)(26,47,66,86)(27,40,67,91)(28,45,68,96)(29,38,69,89)(30,43,70,94)(31,48,71,87)(32,41,72,92)(33,46,61,85)(34,39,62,90)(35,44,63,95)(36,37,64,88) );
G=PermutationGroup([[(1,43,74,94),(2,95,75,44),(3,45,76,96),(4,85,77,46),(5,47,78,86),(6,87,79,48),(7,37,80,88),(8,89,81,38),(9,39,82,90),(10,91,83,40),(11,41,84,92),(12,93,73,42),(13,63,51,35),(14,36,52,64),(15,65,53,25),(16,26,54,66),(17,67,55,27),(18,28,56,68),(19,69,57,29),(20,30,58,70),(21,71,59,31),(22,32,60,72),(23,61,49,33),(24,34,50,62)], [(1,46,7,40),(2,47,8,41),(3,48,9,42),(4,37,10,43),(5,38,11,44),(6,39,12,45),(13,72,19,66),(14,61,20,67),(15,62,21,68),(16,63,22,69),(17,64,23,70),(18,65,24,71),(25,50,31,56),(26,51,32,57),(27,52,33,58),(28,53,34,59),(29,54,35,60),(30,55,36,49),(73,96,79,90),(74,85,80,91),(75,86,81,92),(76,87,82,93),(77,88,83,94),(78,89,84,95)], [(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)], [(1,20,74,58),(2,13,75,51),(3,18,76,56),(4,23,77,49),(5,16,78,54),(6,21,79,59),(7,14,80,52),(8,19,81,57),(9,24,82,50),(10,17,83,55),(11,22,84,60),(12,15,73,53),(25,42,65,93),(26,47,66,86),(27,40,67,91),(28,45,68,96),(29,38,69,89),(30,43,70,94),(31,48,71,87),(32,41,72,92),(33,46,61,85),(34,39,62,90),(35,44,63,95),(36,37,64,88)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4S | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 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 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- 1+4 | D4⋊2S3 | Q8○D12 |
| kernel | C42.106D6 | C4×Dic6 | C12⋊2Q8 | C23.8D6 | Dic6⋊C4 | C12.48D4 | C23.26D6 | C23.12D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.106D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 5 | 12 |
| 5 | 4 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 1 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 10 | 5 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,5,0,0,10,12],[5,0,0,0,4,8,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,1,11,0,0,0,0,12,0,0,0,3,1],[10,1,0,0,5,3,0,0,0,0,8,0,0,0,2,5] >;
C42.106D6 in GAP, Magma, Sage, TeX
C_4^2._{106}D_6 % in TeX
G:=Group("C4^2.106D6"); // GroupNames label
G:=SmallGroup(192,1101);
// by ID
G=gap.SmallGroup(192,1101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations