metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.163D6, C6.1402+ (1+4), (C4×D12)⋊16C2, C4⋊C4.214D6, C12⋊D4⋊37C2, Dic3⋊D4⋊45C2, C42⋊2C2⋊6S3, C42⋊2S3⋊8C2, C22⋊C4.81D6, Dic3⋊5D4⋊42C2, D6.33(C4○D4), D6.D4⋊42C2, C2.65(D4○D12), Dic3.Q8⋊39C2, (C4×C12).35C22, (C2×C12).96C23, (C2×C6).253C24, D6⋊C4.46C22, Dic3⋊4D4⋊37C2, C23.69(C22×S3), (C22×C6).67C23, Dic3.33(C4○D4), (C2×D12).169C22, C22.D12⋊30C2, C4⋊Dic3.318C22, C22.274(S3×C23), Dic3⋊C4.147C22, (C22×S3).112C23, (C2×Dic3).266C23, (C4×Dic3).152C22, C3⋊11(C22.47C24), (C22×Dic3).153C22, (S3×C4⋊C4)⋊43C2, C4⋊C4⋊S3⋊43C2, C2.100(S3×C4○D4), C6.211(C2×C4○D4), (C3×C42⋊2C2)⋊8C2, (S3×C2×C4).135C22, (C2×C4).89(C22×S3), (C3×C4⋊C4).205C22, (C2×C3⋊D4).73C22, (C3×C22⋊C4).78C22, SmallGroup(192,1268)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 640 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×12], C22, C22 [×13], S3 [×4], C6 [×3], C6, C2×C4 [×6], C2×C4 [×13], D4 [×10], C23, C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×6], D6 [×2], D6 [×8], C2×C6, C2×C6 [×3], C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×6], C2×D4 [×6], C4×S3 [×7], D12 [×5], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×5], C2×C12 [×6], C22×S3 [×3], C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2, C42⋊2C2, C4×Dic3 [×2], Dic3⋊C4 [×5], C4⋊Dic3 [×2], D6⋊C4 [×7], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×5], C2×D12 [×3], C22×Dic3, C2×C3⋊D4 [×3], C22.47C24, C42⋊2S3, C4×D12, Dic3⋊4D4 [×2], Dic3⋊D4 [×3], C22.D12, Dic3.Q8, S3×C4⋊C4, Dic3⋊5D4, D6.D4, C12⋊D4, C4⋊C4⋊S3, C3×C42⋊2C2, C42.163D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), S3×C23, C22.47C24, S3×C4○D4 [×2], D4○D12, C42.163D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 79 7 73)(2 13 8 19)(3 81 9 75)(4 15 10 21)(5 83 11 77)(6 17 12 23)(14 27 20 33)(16 29 22 35)(18 31 24 25)(26 74 32 80)(28 76 34 82)(30 78 36 84)(37 94 43 88)(38 71 44 65)(39 96 45 90)(40 61 46 67)(41 86 47 92)(42 63 48 69)(49 68 55 62)(50 93 56 87)(51 70 57 64)(52 95 58 89)(53 72 59 66)(54 85 60 91)
(1 46 25 54)(2 41 26 49)(3 48 27 56)(4 43 28 51)(5 38 29 58)(6 45 30 53)(7 40 31 60)(8 47 32 55)(9 42 33 50)(10 37 34 57)(11 44 35 52)(12 39 36 59)(13 86 74 68)(14 93 75 63)(15 88 76 70)(16 95 77 65)(17 90 78 72)(18 85 79 67)(19 92 80 62)(20 87 81 69)(21 94 82 64)(22 89 83 71)(23 96 84 66)(24 91 73 61)
(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 50 7 56)(2 55 8 49)(3 60 9 54)(4 53 10 59)(5 58 11 52)(6 51 12 57)(13 86 19 92)(14 91 20 85)(15 96 21 90)(16 89 22 95)(17 94 23 88)(18 87 24 93)(25 42 31 48)(26 47 32 41)(27 40 33 46)(28 45 34 39)(29 38 35 44)(30 43 36 37)(61 81 67 75)(62 74 68 80)(63 79 69 73)(64 84 70 78)(65 77 71 83)(66 82 72 76)
G:=sub<Sym(96)| (1,79,7,73)(2,13,8,19)(3,81,9,75)(4,15,10,21)(5,83,11,77)(6,17,12,23)(14,27,20,33)(16,29,22,35)(18,31,24,25)(26,74,32,80)(28,76,34,82)(30,78,36,84)(37,94,43,88)(38,71,44,65)(39,96,45,90)(40,61,46,67)(41,86,47,92)(42,63,48,69)(49,68,55,62)(50,93,56,87)(51,70,57,64)(52,95,58,89)(53,72,59,66)(54,85,60,91), (1,46,25,54)(2,41,26,49)(3,48,27,56)(4,43,28,51)(5,38,29,58)(6,45,30,53)(7,40,31,60)(8,47,32,55)(9,42,33,50)(10,37,34,57)(11,44,35,52)(12,39,36,59)(13,86,74,68)(14,93,75,63)(15,88,76,70)(16,95,77,65)(17,90,78,72)(18,85,79,67)(19,92,80,62)(20,87,81,69)(21,94,82,64)(22,89,83,71)(23,96,84,66)(24,91,73,61), (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), (1,50,7,56)(2,55,8,49)(3,60,9,54)(4,53,10,59)(5,58,11,52)(6,51,12,57)(13,86,19,92)(14,91,20,85)(15,96,21,90)(16,89,22,95)(17,94,23,88)(18,87,24,93)(25,42,31,48)(26,47,32,41)(27,40,33,46)(28,45,34,39)(29,38,35,44)(30,43,36,37)(61,81,67,75)(62,74,68,80)(63,79,69,73)(64,84,70,78)(65,77,71,83)(66,82,72,76)>;
G:=Group( (1,79,7,73)(2,13,8,19)(3,81,9,75)(4,15,10,21)(5,83,11,77)(6,17,12,23)(14,27,20,33)(16,29,22,35)(18,31,24,25)(26,74,32,80)(28,76,34,82)(30,78,36,84)(37,94,43,88)(38,71,44,65)(39,96,45,90)(40,61,46,67)(41,86,47,92)(42,63,48,69)(49,68,55,62)(50,93,56,87)(51,70,57,64)(52,95,58,89)(53,72,59,66)(54,85,60,91), (1,46,25,54)(2,41,26,49)(3,48,27,56)(4,43,28,51)(5,38,29,58)(6,45,30,53)(7,40,31,60)(8,47,32,55)(9,42,33,50)(10,37,34,57)(11,44,35,52)(12,39,36,59)(13,86,74,68)(14,93,75,63)(15,88,76,70)(16,95,77,65)(17,90,78,72)(18,85,79,67)(19,92,80,62)(20,87,81,69)(21,94,82,64)(22,89,83,71)(23,96,84,66)(24,91,73,61), (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), (1,50,7,56)(2,55,8,49)(3,60,9,54)(4,53,10,59)(5,58,11,52)(6,51,12,57)(13,86,19,92)(14,91,20,85)(15,96,21,90)(16,89,22,95)(17,94,23,88)(18,87,24,93)(25,42,31,48)(26,47,32,41)(27,40,33,46)(28,45,34,39)(29,38,35,44)(30,43,36,37)(61,81,67,75)(62,74,68,80)(63,79,69,73)(64,84,70,78)(65,77,71,83)(66,82,72,76) );
G=PermutationGroup([(1,79,7,73),(2,13,8,19),(3,81,9,75),(4,15,10,21),(5,83,11,77),(6,17,12,23),(14,27,20,33),(16,29,22,35),(18,31,24,25),(26,74,32,80),(28,76,34,82),(30,78,36,84),(37,94,43,88),(38,71,44,65),(39,96,45,90),(40,61,46,67),(41,86,47,92),(42,63,48,69),(49,68,55,62),(50,93,56,87),(51,70,57,64),(52,95,58,89),(53,72,59,66),(54,85,60,91)], [(1,46,25,54),(2,41,26,49),(3,48,27,56),(4,43,28,51),(5,38,29,58),(6,45,30,53),(7,40,31,60),(8,47,32,55),(9,42,33,50),(10,37,34,57),(11,44,35,52),(12,39,36,59),(13,86,74,68),(14,93,75,63),(15,88,76,70),(16,95,77,65),(17,90,78,72),(18,85,79,67),(19,92,80,62),(20,87,81,69),(21,94,82,64),(22,89,83,71),(23,96,84,66),(24,91,73,61)], [(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)], [(1,50,7,56),(2,55,8,49),(3,60,9,54),(4,53,10,59),(5,58,11,52),(6,51,12,57),(13,86,19,92),(14,91,20,85),(15,96,21,90),(16,89,22,95),(17,94,23,88),(18,87,24,93),(25,42,31,48),(26,47,32,41),(27,40,33,46),(28,45,34,39),(29,38,35,44),(30,43,36,37),(61,81,67,75),(62,74,68,80),(63,79,69,73),(64,84,70,78),(65,77,71,83),(66,82,72,76)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,1,0,0,0,0,0,8,0,0,0,0,0,0,5,12,0,0,0,0,11,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,5,0,0,0,0,0,11,8],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2+ (1+4) | S3×C4○D4 | D4○D12 |
| kernel | C42.163D6 | C42⋊2S3 | C4×D12 | Dic3⋊4D4 | Dic3⋊D4 | C22.D12 | Dic3.Q8 | S3×C4⋊C4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{163}D_6 % in TeX
G:=Group("C4^2.163D6"); // GroupNames label
G:=SmallGroup(192,1268);
// by ID
G=gap.SmallGroup(192,1268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,1571,297,192,6278]);
// 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=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations