metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.150D6, C6.282- (1+4), C6.1322+ (1+4), (C4×D12)⋊47C2, C4⋊C4.207D6, C42.C2⋊6S3, C42⋊3S3⋊9C2, D6⋊Q8⋊35C2, C4.D12⋊37C2, Dic3⋊5D4⋊36C2, D6.11(C4○D4), D6.D4⋊34C2, C12⋊D4.12C2, C2.57(D4○D12), Dic3.Q8⋊33C2, (C2×C6).236C24, D6⋊C4.10C22, (C4×C12).196C22, (C2×C12).188C23, (C2×D12).164C22, Dic3⋊C4.52C22, C4⋊Dic3.314C22, C22.257(S3×C23), (C22×S3).102C23, C2.29(Q8.15D6), C3⋊8(C22.33C24), (C2×Dic3).258C23, (C2×Dic6).180C22, (C4×Dic3).143C22, (S3×C4⋊C4)⋊36C2, C4⋊C4⋊S3⋊34C2, C2.87(S3×C4○D4), C6.198(C2×C4○D4), (C3×C42.C2)⋊9C2, (S3×C2×C4).126C22, (C2×C4).80(C22×S3), (C3×C4⋊C4).191C22, SmallGroup(192,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 576 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×10], S3 [×4], C6 [×3], C2×C4 [×7], C2×C4 [×11], D4 [×5], Q8, C23 [×3], Dic3 [×5], C12 [×7], D6 [×2], D6 [×8], C2×C6, C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×8], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic6, C4×S3 [×6], D12 [×5], C2×Dic3 [×5], C2×C12 [×7], C22×S3 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2, C42.C2, C42⋊2C2 [×2], C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×6], C2×Dic6, S3×C2×C4 [×5], C2×D12 [×3], C22.33C24, C4×D12, C42⋊3S3, Dic3.Q8, S3×C4⋊C4, Dic3⋊5D4, D6.D4 [×4], C12⋊D4, D6⋊Q8 [×2], C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.150D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), S3×C23, C22.33C24, Q8.15D6, S3×C4○D4, D4○D12, C42.150D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 81 47 68)(2 76 48 63)(3 83 37 70)(4 78 38 65)(5 73 39 72)(6 80 40 67)(7 75 41 62)(8 82 42 69)(9 77 43 64)(10 84 44 71)(11 79 45 66)(12 74 46 61)(13 35 90 49)(14 30 91 56)(15 25 92 51)(16 32 93 58)(17 27 94 53)(18 34 95 60)(19 29 96 55)(20 36 85 50)(21 31 86 57)(22 26 87 52)(23 33 88 59)(24 28 89 54)
(1 33 41 53)(2 54 42 34)(3 35 43 55)(4 56 44 36)(5 25 45 57)(6 58 46 26)(7 27 47 59)(8 60 48 28)(9 29 37 49)(10 50 38 30)(11 31 39 51)(12 52 40 32)(13 77 96 70)(14 71 85 78)(15 79 86 72)(16 61 87 80)(17 81 88 62)(18 63 89 82)(19 83 90 64)(20 65 91 84)(21 73 92 66)(22 67 93 74)(23 75 94 68)(24 69 95 76)
(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 87 7 93)(2 92 8 86)(3 85 9 91)(4 90 10 96)(5 95 11 89)(6 88 12 94)(13 44 19 38)(14 37 20 43)(15 42 21 48)(16 47 22 41)(17 40 23 46)(18 45 24 39)(25 63 31 69)(26 68 32 62)(27 61 33 67)(28 66 34 72)(29 71 35 65)(30 64 36 70)(49 78 55 84)(50 83 56 77)(51 76 57 82)(52 81 58 75)(53 74 59 80)(54 79 60 73)
G:=sub<Sym(96)| (1,81,47,68)(2,76,48,63)(3,83,37,70)(4,78,38,65)(5,73,39,72)(6,80,40,67)(7,75,41,62)(8,82,42,69)(9,77,43,64)(10,84,44,71)(11,79,45,66)(12,74,46,61)(13,35,90,49)(14,30,91,56)(15,25,92,51)(16,32,93,58)(17,27,94,53)(18,34,95,60)(19,29,96,55)(20,36,85,50)(21,31,86,57)(22,26,87,52)(23,33,88,59)(24,28,89,54), (1,33,41,53)(2,54,42,34)(3,35,43,55)(4,56,44,36)(5,25,45,57)(6,58,46,26)(7,27,47,59)(8,60,48,28)(9,29,37,49)(10,50,38,30)(11,31,39,51)(12,52,40,32)(13,77,96,70)(14,71,85,78)(15,79,86,72)(16,61,87,80)(17,81,88,62)(18,63,89,82)(19,83,90,64)(20,65,91,84)(21,73,92,66)(22,67,93,74)(23,75,94,68)(24,69,95,76), (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), (1,87,7,93)(2,92,8,86)(3,85,9,91)(4,90,10,96)(5,95,11,89)(6,88,12,94)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,63,31,69)(26,68,32,62)(27,61,33,67)(28,66,34,72)(29,71,35,65)(30,64,36,70)(49,78,55,84)(50,83,56,77)(51,76,57,82)(52,81,58,75)(53,74,59,80)(54,79,60,73)>;
G:=Group( (1,81,47,68)(2,76,48,63)(3,83,37,70)(4,78,38,65)(5,73,39,72)(6,80,40,67)(7,75,41,62)(8,82,42,69)(9,77,43,64)(10,84,44,71)(11,79,45,66)(12,74,46,61)(13,35,90,49)(14,30,91,56)(15,25,92,51)(16,32,93,58)(17,27,94,53)(18,34,95,60)(19,29,96,55)(20,36,85,50)(21,31,86,57)(22,26,87,52)(23,33,88,59)(24,28,89,54), (1,33,41,53)(2,54,42,34)(3,35,43,55)(4,56,44,36)(5,25,45,57)(6,58,46,26)(7,27,47,59)(8,60,48,28)(9,29,37,49)(10,50,38,30)(11,31,39,51)(12,52,40,32)(13,77,96,70)(14,71,85,78)(15,79,86,72)(16,61,87,80)(17,81,88,62)(18,63,89,82)(19,83,90,64)(20,65,91,84)(21,73,92,66)(22,67,93,74)(23,75,94,68)(24,69,95,76), (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), (1,87,7,93)(2,92,8,86)(3,85,9,91)(4,90,10,96)(5,95,11,89)(6,88,12,94)(13,44,19,38)(14,37,20,43)(15,42,21,48)(16,47,22,41)(17,40,23,46)(18,45,24,39)(25,63,31,69)(26,68,32,62)(27,61,33,67)(28,66,34,72)(29,71,35,65)(30,64,36,70)(49,78,55,84)(50,83,56,77)(51,76,57,82)(52,81,58,75)(53,74,59,80)(54,79,60,73) );
G=PermutationGroup([(1,81,47,68),(2,76,48,63),(3,83,37,70),(4,78,38,65),(5,73,39,72),(6,80,40,67),(7,75,41,62),(8,82,42,69),(9,77,43,64),(10,84,44,71),(11,79,45,66),(12,74,46,61),(13,35,90,49),(14,30,91,56),(15,25,92,51),(16,32,93,58),(17,27,94,53),(18,34,95,60),(19,29,96,55),(20,36,85,50),(21,31,86,57),(22,26,87,52),(23,33,88,59),(24,28,89,54)], [(1,33,41,53),(2,54,42,34),(3,35,43,55),(4,56,44,36),(5,25,45,57),(6,58,46,26),(7,27,47,59),(8,60,48,28),(9,29,37,49),(10,50,38,30),(11,31,39,51),(12,52,40,32),(13,77,96,70),(14,71,85,78),(15,79,86,72),(16,61,87,80),(17,81,88,62),(18,63,89,82),(19,83,90,64),(20,65,91,84),(21,73,92,66),(22,67,93,74),(23,75,94,68),(24,69,95,76)], [(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)], [(1,87,7,93),(2,92,8,86),(3,85,9,91),(4,90,10,96),(5,95,11,89),(6,88,12,94),(13,44,19,38),(14,37,20,43),(15,42,21,48),(16,47,22,41),(17,40,23,46),(18,45,24,39),(25,63,31,69),(26,68,32,62),(27,61,33,67),(28,66,34,72),(29,71,35,65),(30,64,36,70),(49,78,55,84),(50,83,56,77),(51,76,57,82),(52,81,58,75),(53,74,59,80),(54,79,60,73)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 5 | 7 | 6 |
| 0 | 0 | 8 | 3 | 7 | 1 |
| 0 | 0 | 7 | 6 | 5 | 8 |
| 0 | 0 | 7 | 1 | 5 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 9 | 9 |
| 0 | 0 | 0 | 3 | 0 | 4 |
| 0 | 0 | 9 | 9 | 3 | 3 |
| 0 | 0 | 0 | 4 | 0 | 10 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,8,7,7,0,0,5,3,6,1,0,0,7,7,5,5,0,0,6,1,8,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,0,9,0,0,0,10,3,9,4,0,0,9,0,3,0,0,0,9,4,3,10] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2+ (1+4) | 2- (1+4) | Q8.15D6 | S3×C4○D4 | D4○D12 |
| kernel | C42.150D6 | C4×D12 | C42⋊3S3 | Dic3.Q8 | S3×C4⋊C4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | D6 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 6 | 4 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{150}D_6 % in TeX
G:=Group("C4^2.150D6"); // GroupNames label
G:=SmallGroup(192,1251);
// by ID
G=gap.SmallGroup(192,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,570,136,6278]);
// 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=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