metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.165D6, C6.1432+ 1+4, C6.1032- 1+4, C12⋊Q8⋊42C2, C4⋊C4.120D6, C12⋊2Q8⋊9C2, C42⋊3S3⋊2C2, C42⋊2C2⋊9S3, C4.D12⋊43C2, D6⋊Q8⋊46C2, (C4×C12).9C22, C22⋊C4.42D6, C2.68(D4○D12), Dic3.Q8⋊40C2, (C2×C6).256C24, (C2×C12).97C23, D6⋊C4.48C22, C2.67(Q8○D12), C4.Dic6⋊41C2, C23.9D6.5C2, C23.8D6⋊47C2, C4⋊Dic3.55C22, C23.72(C22×S3), (C22×C6).70C23, Dic3.D4⋊48C2, Dic3⋊C4.11C22, C22.277(S3×C23), C23.11D6.5C2, (C2×Dic6).43C22, C23.21D6.3C2, (C22×S3).115C23, C3⋊5(C22.57C24), (C2×Dic3).132C23, (C4×Dic3).153C22, C6.D4.70C22, (C22×Dic3).155C22, C4⋊C4⋊S3⋊45C2, (S3×C2×C4).137C22, (C3×C42⋊2C2)⋊11C2, (C3×C4⋊C4).207C22, (C2×C4).212(C22×S3), (C2×C3⋊D4).76C22, (C3×C22⋊C4).81C22, SmallGroup(192,1271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.165D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
Subgroups: 480 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22.57C24, C12⋊2Q8, C42⋊3S3, Dic3.D4, C23.8D6, C23.9D6, C23.11D6, C23.21D6, C12⋊Q8, Dic3.Q8, C4.Dic6, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.165D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, D4○D12, Q8○D12, C42.165D6
►On 96 points
(1 63 93 60)(2 55 94 70)(3 65 95 50)(4 57 96 72)(5 67 85 52)(6 59 86 62)(7 69 87 54)(8 49 88 64)(9 71 89 56)(10 51 90 66)(11 61 91 58)(12 53 92 68)(13 82 43 26)(14 33 44 77)(15 84 45 28)(16 35 46 79)(17 74 47 30)(18 25 48 81)(19 76 37 32)(20 27 38 83)(21 78 39 34)(22 29 40 73)(23 80 41 36)(24 31 42 75)
(1 84 87 34)(2 79 88 29)(3 74 89 36)(4 81 90 31)(5 76 91 26)(6 83 92 33)(7 78 93 28)(8 73 94 35)(9 80 95 30)(10 75 96 25)(11 82 85 32)(12 77 86 27)(13 67 37 58)(14 62 38 53)(15 69 39 60)(16 64 40 55)(17 71 41 50)(18 66 42 57)(19 61 43 52)(20 68 44 59)(21 63 45 54)(22 70 46 49)(23 65 47 56)(24 72 48 51)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)(25 74 31 80)(26 79 32 73)(27 84 33 78)(28 77 34 83)(29 82 35 76)(30 75 36 81)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 64 67 70)(62 69 68 63)(65 72 71 66)(85 94 91 88)(86 87 92 93)(89 90 95 96)
G:=sub<Sym(96)| (1,63,93,60)(2,55,94,70)(3,65,95,50)(4,57,96,72)(5,67,85,52)(6,59,86,62)(7,69,87,54)(8,49,88,64)(9,71,89,56)(10,51,90,66)(11,61,91,58)(12,53,92,68)(13,82,43,26)(14,33,44,77)(15,84,45,28)(16,35,46,79)(17,74,47,30)(18,25,48,81)(19,76,37,32)(20,27,38,83)(21,78,39,34)(22,29,40,73)(23,80,41,36)(24,31,42,75), (1,84,87,34)(2,79,88,29)(3,74,89,36)(4,81,90,31)(5,76,91,26)(6,83,92,33)(7,78,93,28)(8,73,94,35)(9,80,95,30)(10,75,96,25)(11,82,85,32)(12,77,86,27)(13,67,37,58)(14,62,38,53)(15,69,39,60)(16,64,40,55)(17,71,41,50)(18,66,42,57)(19,61,43,52)(20,68,44,59)(21,63,45,54)(22,70,46,49)(23,65,47,56)(24,72,48,51), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41)(25,74,31,80)(26,79,32,73)(27,84,33,78)(28,77,34,83)(29,82,35,76)(30,75,36,81)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,94,91,88)(86,87,92,93)(89,90,95,96)>;
G:=Group( (1,63,93,60)(2,55,94,70)(3,65,95,50)(4,57,96,72)(5,67,85,52)(6,59,86,62)(7,69,87,54)(8,49,88,64)(9,71,89,56)(10,51,90,66)(11,61,91,58)(12,53,92,68)(13,82,43,26)(14,33,44,77)(15,84,45,28)(16,35,46,79)(17,74,47,30)(18,25,48,81)(19,76,37,32)(20,27,38,83)(21,78,39,34)(22,29,40,73)(23,80,41,36)(24,31,42,75), (1,84,87,34)(2,79,88,29)(3,74,89,36)(4,81,90,31)(5,76,91,26)(6,83,92,33)(7,78,93,28)(8,73,94,35)(9,80,95,30)(10,75,96,25)(11,82,85,32)(12,77,86,27)(13,67,37,58)(14,62,38,53)(15,69,39,60)(16,64,40,55)(17,71,41,50)(18,66,42,57)(19,61,43,52)(20,68,44,59)(21,63,45,54)(22,70,46,49)(23,65,47,56)(24,72,48,51), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41)(25,74,31,80)(26,79,32,73)(27,84,33,78)(28,77,34,83)(29,82,35,76)(30,75,36,81)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,94,91,88)(86,87,92,93)(89,90,95,96) );
G=PermutationGroup([[(1,63,93,60),(2,55,94,70),(3,65,95,50),(4,57,96,72),(5,67,85,52),(6,59,86,62),(7,69,87,54),(8,49,88,64),(9,71,89,56),(10,51,90,66),(11,61,91,58),(12,53,92,68),(13,82,43,26),(14,33,44,77),(15,84,45,28),(16,35,46,79),(17,74,47,30),(18,25,48,81),(19,76,37,32),(20,27,38,83),(21,78,39,34),(22,29,40,73),(23,80,41,36),(24,31,42,75)], [(1,84,87,34),(2,79,88,29),(3,74,89,36),(4,81,90,31),(5,76,91,26),(6,83,92,33),(7,78,93,28),(8,73,94,35),(9,80,95,30),(10,75,96,25),(11,82,85,32),(12,77,86,27),(13,67,37,58),(14,62,38,53),(15,69,39,60),(16,64,40,55),(17,71,41,50),(18,66,42,57),(19,61,43,52),(20,68,44,59),(21,63,45,54),(22,70,46,49),(23,65,47,56),(24,72,48,51)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,46,19,40),(14,39,20,45),(15,44,21,38),(16,37,22,43),(17,42,23,48),(18,47,24,41),(25,74,31,80),(26,79,32,73),(27,84,33,78),(28,77,34,83),(29,82,35,76),(30,75,36,81),(49,52,55,58),(50,57,56,51),(53,60,59,54),(61,64,67,70),(62,69,68,63),(65,72,71,66),(85,94,91,88),(86,87,92,93),(89,90,95,96)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | ··· | 4M | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | D4○D12 | Q8○D12 |
| kernel | C42.165D6 | C12⋊2Q8 | C42⋊3S3 | Dic3.D4 | C23.8D6 | C23.9D6 | C23.11D6 | C23.21D6 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 2 | 2 | 4 |
Matrix representation of C42.165D6 ►in GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 4 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 4 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,10,4,0,0],[10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,10,4],[0,0,8,0,0,0,0,0,0,0,8,5,0,0,0,0,5,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9] >;
C42.165D6 in GAP, Magma, Sage, TeX
C_4^2._{165}D_6 % in TeX
G:=Group("C4^2.165D6"); // GroupNames label
G:=SmallGroup(192,1271);
// by ID
G=gap.SmallGroup(192,1271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations