metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.158D6, C6.322- 1+4, C6.1362+ 1+4, C4⋊C4.115D6, C12⋊D4⋊35C2, D6⋊Q8⋊38C2, C42.C2⋊14S3, C42⋊7S3⋊32C2, D6.D4⋊37C2, C2.61(D4○D12), (C2×C6).244C24, D6⋊C4.74C22, (C4×C12).225C22, (C2×C12).191C23, (C2×D12).167C22, Dic3⋊C4.55C22, C22.265(S3×C23), (C2×Dic6).42C22, (C22×S3).109C23, C2.33(Q8.15D6), C3⋊5(C22.56C24), (C2×Dic3).126C23, (S3×C2×C4).134C22, (C3×C42.C2)⋊17C2, (C3×C4⋊C4).199C22, (C2×C4).208(C22×S3), SmallGroup(192,1259)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.158D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 656 in 220 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, Dic3⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22.56C24, C42⋊7S3, D6.D4, C12⋊D4, D6⋊Q8, C3×C42.C2, C42.158D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.56C24, Q8.15D6, D4○D12, C42.158D6
►On 96 points
Generators in S96
(1 17 7 23)(2 35 8 29)(3 19 9 13)(4 25 10 31)(5 21 11 15)(6 27 12 33)(14 60 20 54)(16 50 22 56)(18 52 24 58)(26 55 32 49)(28 57 34 51)(30 59 36 53)(37 65 43 71)(38 89 44 95)(39 67 45 61)(40 91 46 85)(41 69 47 63)(42 93 48 87)(62 77 68 83)(64 79 70 73)(66 81 72 75)(74 88 80 94)(76 90 82 96)(78 92 84 86)
(1 74 51 43)(2 44 52 75)(3 76 53 45)(4 46 54 77)(5 78 55 47)(6 48 56 79)(7 80 57 37)(8 38 58 81)(9 82 59 39)(10 40 60 83)(11 84 49 41)(12 42 50 73)(13 96 36 67)(14 68 25 85)(15 86 26 69)(16 70 27 87)(17 88 28 71)(18 72 29 89)(19 90 30 61)(20 62 31 91)(21 92 32 63)(22 64 33 93)(23 94 34 65)(24 66 35 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 39 7 45)(2 44 8 38)(3 37 9 43)(4 42 10 48)(5 47 11 41)(6 40 12 46)(13 65 19 71)(14 70 20 64)(15 63 21 69)(16 68 22 62)(17 61 23 67)(18 66 24 72)(25 87 31 93)(26 92 32 86)(27 85 33 91)(28 90 34 96)(29 95 35 89)(30 88 36 94)(49 84 55 78)(50 77 56 83)(51 82 57 76)(52 75 58 81)(53 80 59 74)(54 73 60 79)
G:=sub<Sym(96)| (1,17,7,23)(2,35,8,29)(3,19,9,13)(4,25,10,31)(5,21,11,15)(6,27,12,33)(14,60,20,54)(16,50,22,56)(18,52,24,58)(26,55,32,49)(28,57,34,51)(30,59,36,53)(37,65,43,71)(38,89,44,95)(39,67,45,61)(40,91,46,85)(41,69,47,63)(42,93,48,87)(62,77,68,83)(64,79,70,73)(66,81,72,75)(74,88,80,94)(76,90,82,96)(78,92,84,86), (1,74,51,43)(2,44,52,75)(3,76,53,45)(4,46,54,77)(5,78,55,47)(6,48,56,79)(7,80,57,37)(8,38,58,81)(9,82,59,39)(10,40,60,83)(11,84,49,41)(12,42,50,73)(13,96,36,67)(14,68,25,85)(15,86,26,69)(16,70,27,87)(17,88,28,71)(18,72,29,89)(19,90,30,61)(20,62,31,91)(21,92,32,63)(22,64,33,93)(23,94,34,65)(24,66,35,95), (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), (1,39,7,45)(2,44,8,38)(3,37,9,43)(4,42,10,48)(5,47,11,41)(6,40,12,46)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(25,87,31,93)(26,92,32,86)(27,85,33,91)(28,90,34,96)(29,95,35,89)(30,88,36,94)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79)>;
G:=Group( (1,17,7,23)(2,35,8,29)(3,19,9,13)(4,25,10,31)(5,21,11,15)(6,27,12,33)(14,60,20,54)(16,50,22,56)(18,52,24,58)(26,55,32,49)(28,57,34,51)(30,59,36,53)(37,65,43,71)(38,89,44,95)(39,67,45,61)(40,91,46,85)(41,69,47,63)(42,93,48,87)(62,77,68,83)(64,79,70,73)(66,81,72,75)(74,88,80,94)(76,90,82,96)(78,92,84,86), (1,74,51,43)(2,44,52,75)(3,76,53,45)(4,46,54,77)(5,78,55,47)(6,48,56,79)(7,80,57,37)(8,38,58,81)(9,82,59,39)(10,40,60,83)(11,84,49,41)(12,42,50,73)(13,96,36,67)(14,68,25,85)(15,86,26,69)(16,70,27,87)(17,88,28,71)(18,72,29,89)(19,90,30,61)(20,62,31,91)(21,92,32,63)(22,64,33,93)(23,94,34,65)(24,66,35,95), (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), (1,39,7,45)(2,44,8,38)(3,37,9,43)(4,42,10,48)(5,47,11,41)(6,40,12,46)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(25,87,31,93)(26,92,32,86)(27,85,33,91)(28,90,34,96)(29,95,35,89)(30,88,36,94)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79) );
G=PermutationGroup([[(1,17,7,23),(2,35,8,29),(3,19,9,13),(4,25,10,31),(5,21,11,15),(6,27,12,33),(14,60,20,54),(16,50,22,56),(18,52,24,58),(26,55,32,49),(28,57,34,51),(30,59,36,53),(37,65,43,71),(38,89,44,95),(39,67,45,61),(40,91,46,85),(41,69,47,63),(42,93,48,87),(62,77,68,83),(64,79,70,73),(66,81,72,75),(74,88,80,94),(76,90,82,96),(78,92,84,86)], [(1,74,51,43),(2,44,52,75),(3,76,53,45),(4,46,54,77),(5,78,55,47),(6,48,56,79),(7,80,57,37),(8,38,58,81),(9,82,59,39),(10,40,60,83),(11,84,49,41),(12,42,50,73),(13,96,36,67),(14,68,25,85),(15,86,26,69),(16,70,27,87),(17,88,28,71),(18,72,29,89),(19,90,30,61),(20,62,31,91),(21,92,32,63),(22,64,33,93),(23,94,34,65),(24,66,35,95)], [(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)], [(1,39,7,45),(2,44,8,38),(3,37,9,43),(4,42,10,48),(5,47,11,41),(6,40,12,46),(13,65,19,71),(14,70,20,64),(15,63,21,69),(16,68,22,62),(17,61,23,67),(18,66,24,72),(25,87,31,93),(26,92,32,86),(27,85,33,91),(28,90,34,96),(29,95,35,89),(30,88,36,94),(49,84,55,78),(50,77,56,83),(51,82,57,76),(52,75,58,81),(53,80,59,74),(54,73,60,79)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2+ 1+4 | 2- 1+4 | Q8.15D6 | D4○D12 |
| kernel | C42.158D6 | C42⋊7S3 | D6.D4 | C12⋊D4 | D6⋊Q8 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 6 | 2 | 1 | 2 | 4 |
Matrix representation of C42.158D6 ►in GL8(𝔽13)
| 3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 7 | 6 | 5 | 8 | 0 | 0 | 0 | 0 |
| 7 | 1 | 5 | 10 | 0 | 0 | 0 | 0 |
| 5 | 8 | 6 | 7 | 0 | 0 | 0 | 0 |
| 5 | 10 | 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 2 | 11 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 12 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 12 | 11 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(13))| [3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0],[0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[7,7,5,5,0,0,0,0,6,1,8,10,0,0,0,0,5,5,6,6,0,0,0,0,8,10,7,12,0,0,0,0,0,0,0,0,1,1,2,11,0,0,0,0,12,2,2,0,0,0,0,0,2,11,12,12,0,0,0,0,2,0,1,11],[0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0] >;
C42.158D6 in GAP, Magma, Sage, TeX
C_4^2._{158}D_6 % in TeX
G:=Group("C4^2.158D6"); // GroupNames label
G:=SmallGroup(192,1259);
// by ID
G=gap.SmallGroup(192,1259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations