metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.180D6, C6.862+ 1+4, C6.412- 1+4, C4⋊Q8⋊18S3, C4⋊C4.127D6, D6⋊3Q8⋊39C2, D6⋊Q8⋊49C2, (C2×Q8).114D6, C42⋊3S3⋊20C2, Dic3.Q8⋊43C2, (C2×C6).279C24, D6⋊C4.76C22, D6.D4.5C2, Dic3⋊Q8⋊28C2, C2.90(D4⋊6D6), (C2×C12).641C23, (C4×C12).274C22, C12.23D4.9C2, (C6×Q8).146C22, (C2×D12).174C22, Dic3⋊C4.88C22, C4⋊Dic3.256C22, C22.300(S3×C23), (C22×S3).124C23, C2.42(Q8.15D6), C3⋊6(C22.57C24), (C4×Dic3).168C22, (C2×Dic3).147C23, (C2×Dic6).193C22, (C3×C4⋊Q8)⋊21C2, C4⋊C4⋊S3⋊48C2, (S3×C2×C4).152C22, (C3×C4⋊C4).222C22, (C2×C4).222(C22×S3), SmallGroup(192,1294)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.180D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 480 in 196 conjugacy classes, 91 normal (27 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, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, C22.57C24, C42⋊3S3, Dic3.Q8, D6.D4, D6⋊Q8, C4⋊C4⋊S3, Dic3⋊Q8, D6⋊3Q8, C12.23D4, C3×C4⋊Q8, C42.180D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, D4⋊6D6, Q8.15D6, C42.180D6
►On 96 points
Generators in S96
(1 34 7 28)(2 29 8 35)(3 36 9 30)(4 31 10 25)(5 26 11 32)(6 33 12 27)(13 89 19 95)(14 96 20 90)(15 91 21 85)(16 86 22 92)(17 93 23 87)(18 88 24 94)(37 51 43 57)(38 58 44 52)(39 53 45 59)(40 60 46 54)(41 55 47 49)(42 50 48 56)(61 83 67 77)(62 78 68 84)(63 73 69 79)(64 80 70 74)(65 75 71 81)(66 82 72 76)
(1 50 77 96)(2 85 78 51)(3 52 79 86)(4 87 80 53)(5 54 81 88)(6 89 82 55)(7 56 83 90)(8 91 84 57)(9 58 73 92)(10 93 74 59)(11 60 75 94)(12 95 76 49)(13 66 41 27)(14 28 42 67)(15 68 43 29)(16 30 44 69)(17 70 45 31)(18 32 46 71)(19 72 47 33)(20 34 48 61)(21 62 37 35)(22 36 38 63)(23 64 39 25)(24 26 40 65)
(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 20 19 14)(15 18 21 24)(16 23 22 17)(25 69 31 63)(26 62 32 68)(27 67 33 61)(28 72 34 66)(29 65 35 71)(30 70 36 64)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 90 55 96)(50 95 56 89)(51 88 57 94)(52 93 58 87)(53 86 59 92)(54 91 60 85)(73 74 79 80)(75 84 81 78)(76 77 82 83)
G:=sub<Sym(96)| (1,34,7,28)(2,29,8,35)(3,36,9,30)(4,31,10,25)(5,26,11,32)(6,33,12,27)(13,89,19,95)(14,96,20,90)(15,91,21,85)(16,86,22,92)(17,93,23,87)(18,88,24,94)(37,51,43,57)(38,58,44,52)(39,53,45,59)(40,60,46,54)(41,55,47,49)(42,50,48,56)(61,83,67,77)(62,78,68,84)(63,73,69,79)(64,80,70,74)(65,75,71,81)(66,82,72,76), (1,50,77,96)(2,85,78,51)(3,52,79,86)(4,87,80,53)(5,54,81,88)(6,89,82,55)(7,56,83,90)(8,91,84,57)(9,58,73,92)(10,93,74,59)(11,60,75,94)(12,95,76,49)(13,66,41,27)(14,28,42,67)(15,68,43,29)(16,30,44,69)(17,70,45,31)(18,32,46,71)(19,72,47,33)(20,34,48,61)(21,62,37,35)(22,36,38,63)(23,64,39,25)(24,26,40,65), (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)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,69,31,63)(26,62,32,68)(27,67,33,61)(28,72,34,66)(29,65,35,71)(30,70,36,64)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,90,55,96)(50,95,56,89)(51,88,57,94)(52,93,58,87)(53,86,59,92)(54,91,60,85)(73,74,79,80)(75,84,81,78)(76,77,82,83)>;
G:=Group( (1,34,7,28)(2,29,8,35)(3,36,9,30)(4,31,10,25)(5,26,11,32)(6,33,12,27)(13,89,19,95)(14,96,20,90)(15,91,21,85)(16,86,22,92)(17,93,23,87)(18,88,24,94)(37,51,43,57)(38,58,44,52)(39,53,45,59)(40,60,46,54)(41,55,47,49)(42,50,48,56)(61,83,67,77)(62,78,68,84)(63,73,69,79)(64,80,70,74)(65,75,71,81)(66,82,72,76), (1,50,77,96)(2,85,78,51)(3,52,79,86)(4,87,80,53)(5,54,81,88)(6,89,82,55)(7,56,83,90)(8,91,84,57)(9,58,73,92)(10,93,74,59)(11,60,75,94)(12,95,76,49)(13,66,41,27)(14,28,42,67)(15,68,43,29)(16,30,44,69)(17,70,45,31)(18,32,46,71)(19,72,47,33)(20,34,48,61)(21,62,37,35)(22,36,38,63)(23,64,39,25)(24,26,40,65), (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)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,69,31,63)(26,62,32,68)(27,67,33,61)(28,72,34,66)(29,65,35,71)(30,70,36,64)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,90,55,96)(50,95,56,89)(51,88,57,94)(52,93,58,87)(53,86,59,92)(54,91,60,85)(73,74,79,80)(75,84,81,78)(76,77,82,83) );
G=PermutationGroup([[(1,34,7,28),(2,29,8,35),(3,36,9,30),(4,31,10,25),(5,26,11,32),(6,33,12,27),(13,89,19,95),(14,96,20,90),(15,91,21,85),(16,86,22,92),(17,93,23,87),(18,88,24,94),(37,51,43,57),(38,58,44,52),(39,53,45,59),(40,60,46,54),(41,55,47,49),(42,50,48,56),(61,83,67,77),(62,78,68,84),(63,73,69,79),(64,80,70,74),(65,75,71,81),(66,82,72,76)], [(1,50,77,96),(2,85,78,51),(3,52,79,86),(4,87,80,53),(5,54,81,88),(6,89,82,55),(7,56,83,90),(8,91,84,57),(9,58,73,92),(10,93,74,59),(11,60,75,94),(12,95,76,49),(13,66,41,27),(14,28,42,67),(15,68,43,29),(16,30,44,69),(17,70,45,31),(18,32,46,71),(19,72,47,33),(20,34,48,61),(21,62,37,35),(22,36,38,63),(23,64,39,25),(24,26,40,65)], [(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),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,69,31,63),(26,62,32,68),(27,67,33,61),(28,72,34,66),(29,65,35,71),(30,70,36,64),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,90,55,96),(50,95,56,89),(51,88,57,94),(52,93,58,87),(53,86,59,92),(54,91,60,85),(73,74,79,80),(75,84,81,78),(76,77,82,83)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4G | 4H | ··· | 4M | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 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 | S3 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8.15D6 |
| kernel | C42.180D6 | C42⋊3S3 | Dic3.Q8 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | Dic3⋊Q8 | D6⋊3Q8 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of C42.180D6 ►in GL10(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 12 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 12 | 0 | 0 | 0 | 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 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 12 | 0 | 0 | 0 | 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 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,8,0,5,0,0,0,0,0,0,3,1,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,5,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0],[12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,5,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0] >;
C42.180D6 in GAP, Magma, Sage, TeX
C_4^2._{180}D_6 % in TeX
G:=Group("C4^2.180D6"); // GroupNames label
G:=SmallGroup(192,1294);
// by ID
G=gap.SmallGroup(192,1294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,1571,570,136,6278]);
// 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,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations