metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.89D6, C6.472- 1+4, C4⋊C4.267D6, C12⋊2Q8⋊5C2, (C4×Dic6)⋊7C2, Dic3.Q8⋊4C2, (C2×C6).62C24, C22⋊C4.90D6, C2.6(Q8○D12), C12.6Q8⋊3C2, (C4×C12).22C22, (C22×C4).202D6, C12.235(C4○D4), C4.119(C4○D12), (C2×C12).141C23, C42⋊C2.12S3, C4⋊Dic3.31C22, C23.93(C22×S3), C22.95(S3×C23), C23.8D6.1C2, C12.48D4.18C2, Dic3⋊C4.74C22, (C22×C6).132C23, (C2×Dic3).21C23, C6.D4.3C22, (C22×C12).307C22, C3⋊1(C22.35C24), (C2×Dic6).229C22, (C4×Dic3).194C22, C6.27(C2×C4○D4), C2.29(C2×C4○D12), (C3×C4⋊C4).303C22, (C2×C4).269(C22×S3), (C3×C42⋊C2).13C2, (C3×C22⋊C4).111C22, SmallGroup(192,1077)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.89D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c5 >
Subgroups: 392 in 192 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C22.35C24, C4×Dic6, C12⋊2Q8, C12.6Q8, C23.8D6, Dic3.Q8, C12.48D4, C3×C42⋊C2, C42.89D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, S3×C23, C22.35C24, C2×C4○D12, Q8○D12, C42.89D6
►On 96 points
Generators in S96
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 69 43 63)(38 70 44 64)(39 71 45 65)(40 72 46 66)(41 61 47 67)(42 62 48 68)(49 52 55 58)(50 53 56 59)(51 54 57 60)(73 91 79 85)(74 92 80 86)(75 93 81 87)(76 94 82 88)(77 95 83 89)(78 96 84 90)
(1 32 58 18)(2 27 59 13)(3 34 60 20)(4 29 49 15)(5 36 50 22)(6 31 51 17)(7 26 52 24)(8 33 53 19)(9 28 54 14)(10 35 55 21)(11 30 56 16)(12 25 57 23)(37 87 72 84)(38 94 61 79)(39 89 62 74)(40 96 63 81)(41 91 64 76)(42 86 65 83)(43 93 66 78)(44 88 67 73)(45 95 68 80)(46 90 69 75)(47 85 70 82)(48 92 71 77)
(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 43 58 66)(2 71 59 48)(3 41 60 64)(4 69 49 46)(5 39 50 62)(6 67 51 44)(7 37 52 72)(8 65 53 42)(9 47 54 70)(10 63 55 40)(11 45 56 68)(12 61 57 38)(13 86 27 83)(14 76 28 91)(15 96 29 81)(16 74 30 89)(17 94 31 79)(18 84 32 87)(19 92 33 77)(20 82 34 85)(21 90 35 75)(22 80 36 95)(23 88 25 73)(24 78 26 93)
G:=sub<Sym(96)| (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,69,43,63)(38,70,44,64)(39,71,45,65)(40,72,46,66)(41,61,47,67)(42,62,48,68)(49,52,55,58)(50,53,56,59)(51,54,57,60)(73,91,79,85)(74,92,80,86)(75,93,81,87)(76,94,82,88)(77,95,83,89)(78,96,84,90), (1,32,58,18)(2,27,59,13)(3,34,60,20)(4,29,49,15)(5,36,50,22)(6,31,51,17)(7,26,52,24)(8,33,53,19)(9,28,54,14)(10,35,55,21)(11,30,56,16)(12,25,57,23)(37,87,72,84)(38,94,61,79)(39,89,62,74)(40,96,63,81)(41,91,64,76)(42,86,65,83)(43,93,66,78)(44,88,67,73)(45,95,68,80)(46,90,69,75)(47,85,70,82)(48,92,71,77), (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), (1,43,58,66)(2,71,59,48)(3,41,60,64)(4,69,49,46)(5,39,50,62)(6,67,51,44)(7,37,52,72)(8,65,53,42)(9,47,54,70)(10,63,55,40)(11,45,56,68)(12,61,57,38)(13,86,27,83)(14,76,28,91)(15,96,29,81)(16,74,30,89)(17,94,31,79)(18,84,32,87)(19,92,33,77)(20,82,34,85)(21,90,35,75)(22,80,36,95)(23,88,25,73)(24,78,26,93)>;
G:=Group( (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,69,43,63)(38,70,44,64)(39,71,45,65)(40,72,46,66)(41,61,47,67)(42,62,48,68)(49,52,55,58)(50,53,56,59)(51,54,57,60)(73,91,79,85)(74,92,80,86)(75,93,81,87)(76,94,82,88)(77,95,83,89)(78,96,84,90), (1,32,58,18)(2,27,59,13)(3,34,60,20)(4,29,49,15)(5,36,50,22)(6,31,51,17)(7,26,52,24)(8,33,53,19)(9,28,54,14)(10,35,55,21)(11,30,56,16)(12,25,57,23)(37,87,72,84)(38,94,61,79)(39,89,62,74)(40,96,63,81)(41,91,64,76)(42,86,65,83)(43,93,66,78)(44,88,67,73)(45,95,68,80)(46,90,69,75)(47,85,70,82)(48,92,71,77), (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), (1,43,58,66)(2,71,59,48)(3,41,60,64)(4,69,49,46)(5,39,50,62)(6,67,51,44)(7,37,52,72)(8,65,53,42)(9,47,54,70)(10,63,55,40)(11,45,56,68)(12,61,57,38)(13,86,27,83)(14,76,28,91)(15,96,29,81)(16,74,30,89)(17,94,31,79)(18,84,32,87)(19,92,33,77)(20,82,34,85)(21,90,35,75)(22,80,36,95)(23,88,25,73)(24,78,26,93) );
G=PermutationGroup([[(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,69,43,63),(38,70,44,64),(39,71,45,65),(40,72,46,66),(41,61,47,67),(42,62,48,68),(49,52,55,58),(50,53,56,59),(51,54,57,60),(73,91,79,85),(74,92,80,86),(75,93,81,87),(76,94,82,88),(77,95,83,89),(78,96,84,90)], [(1,32,58,18),(2,27,59,13),(3,34,60,20),(4,29,49,15),(5,36,50,22),(6,31,51,17),(7,26,52,24),(8,33,53,19),(9,28,54,14),(10,35,55,21),(11,30,56,16),(12,25,57,23),(37,87,72,84),(38,94,61,79),(39,89,62,74),(40,96,63,81),(41,91,64,76),(42,86,65,83),(43,93,66,78),(44,88,67,73),(45,95,68,80),(46,90,69,75),(47,85,70,82),(48,92,71,77)], [(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)], [(1,43,58,66),(2,71,59,48),(3,41,60,64),(4,69,49,46),(5,39,50,62),(6,67,51,44),(7,37,52,72),(8,65,53,42),(9,47,54,70),(10,63,55,40),(11,45,56,68),(12,61,57,38),(13,86,27,83),(14,76,28,91),(15,96,29,81),(16,74,30,89),(17,94,31,79),(18,84,32,87),(19,92,33,77),(20,82,34,85),(21,90,35,75),(22,80,36,95),(23,88,25,73),(24,78,26,93)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- 1+4 | Q8○D12 |
| kernel | C42.89D6 | C4×Dic6 | C12⋊2Q8 | C12.6Q8 | C23.8D6 | Dic3.Q8 | C12.48D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 2 | 4 |
Matrix representation of C42.89D6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 12 | 3 |
| 0 | 0 | 0 | 0 | 6 | 0 | 8 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 7 | 5 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 7 | 12 | 3 |
| 0 | 0 | 0 | 0 | 1 | 8 | 8 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 1 | 10 |
| 0 | 0 | 0 | 0 | 11 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 5 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,9,6,0,0,0,0,12,0,4,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,7,0,1,0,0,0,0,0,5,0,12,0,0,0,0,5,5,1,0,0,0,0,0,0,11,0,8],[1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,9,1,0,0,0,0,1,0,7,8,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1],[1,3,0,0,0,0,0,0,8,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,4,11,7,0,0,0,0,0,1,0,0,0,0,0,0,12,1,5,5,0,0,0,0,0,10,0,12] >;
C42.89D6 in GAP, Magma, Sage, TeX
C_4^2._{89}D_6 % in TeX
G:=Group("C4^2.89D6"); // GroupNames label
G:=SmallGroup(192,1077);
// by ID
G=gap.SmallGroup(192,1077);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations