metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.277D6, (C4×D12)⋊4C2, (C2×C42)⋊13S3, (C4×Dic6)⋊4C2, (C2×C6).23C24, C42⋊2S3⋊27C2, C42⋊3S3⋊21C2, C42⋊7S3⋊34C2, C12⋊7D4.20C2, D6⋊C4.82C22, C12.6Q8⋊32C2, (C22×C4).426D6, C4.118(C4○D12), C12.234(C4○D4), C12.48D4⋊50C2, (C2×C12).696C23, (C4×C12).316C22, (C22×S3).5C23, C22.66(S3×C23), (C2×Dic3).7C23, (C2×D12).204C22, C22.22(C4○D12), C23.28D6⋊32C2, Dic3⋊C4.96C22, C4⋊Dic3.290C22, (C22×C6).385C23, C23.229(C22×S3), (C22×C12).565C22, C3⋊1(C23.36C23), (C2×Dic6).225C22, (C4×Dic3).191C22, C6.D4.81C22, (C2×C4×C12)⋊14C2, (C4×C3⋊D4)⋊32C2, C6.10(C2×C4○D4), C2.12(C2×C4○D12), (C2×C6).99(C4○D4), (S3×C2×C4).188C22, (C2×C4).651(C22×S3), (C2×C3⋊D4).86C22, SmallGroup(192,1038)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.277D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 536 in 234 conjugacy classes, 103 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C23.36C23, C4×Dic6, C12.6Q8, C42⋊2S3, C4×D12, C42⋊7S3, C42⋊3S3, C12.48D4, C4×C3⋊D4, C23.28D6, C12⋊7D4, C2×C4×C12, C42.277D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C4○D12, S3×C23, C23.36C23, C2×C4○D12, C42.277D6
►On 96 points
(1 30 15 68)(2 25 16 69)(3 26 17 70)(4 27 18 71)(5 28 13 72)(6 29 14 67)(7 19 56 63)(8 20 57 64)(9 21 58 65)(10 22 59 66)(11 23 60 61)(12 24 55 62)(31 90 75 46)(32 85 76 47)(33 86 77 48)(34 87 78 43)(35 88 73 44)(36 89 74 45)(37 96 81 50)(38 91 82 51)(39 92 83 52)(40 93 84 53)(41 94 79 54)(42 95 80 49)
(1 79 73 65)(2 80 74 66)(3 81 75 61)(4 82 76 62)(5 83 77 63)(6 84 78 64)(7 28 52 48)(8 29 53 43)(9 30 54 44)(10 25 49 45)(11 26 50 46)(12 27 51 47)(13 39 33 19)(14 40 34 20)(15 41 35 21)(16 42 36 22)(17 37 31 23)(18 38 32 24)(55 71 91 85)(56 72 92 86)(57 67 93 87)(58 68 94 88)(59 69 95 89)(60 70 96 90)
(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 73 78)(2 77 74 5)(3 4 75 76)(7 95 52 59)(8 58 53 94)(9 93 54 57)(10 56 49 92)(11 91 50 55)(12 60 51 96)(13 16 33 36)(14 35 34 15)(17 18 31 32)(19 66 39 80)(20 79 40 65)(21 64 41 84)(22 83 42 63)(23 62 37 82)(24 81 38 61)(25 28 45 48)(26 47 46 27)(29 30 43 44)(67 68 87 88)(69 72 89 86)(70 85 90 71)
G:=sub<Sym(96)| (1,30,15,68)(2,25,16,69)(3,26,17,70)(4,27,18,71)(5,28,13,72)(6,29,14,67)(7,19,56,63)(8,20,57,64)(9,21,58,65)(10,22,59,66)(11,23,60,61)(12,24,55,62)(31,90,75,46)(32,85,76,47)(33,86,77,48)(34,87,78,43)(35,88,73,44)(36,89,74,45)(37,96,81,50)(38,91,82,51)(39,92,83,52)(40,93,84,53)(41,94,79,54)(42,95,80,49), (1,79,73,65)(2,80,74,66)(3,81,75,61)(4,82,76,62)(5,83,77,63)(6,84,78,64)(7,28,52,48)(8,29,53,43)(9,30,54,44)(10,25,49,45)(11,26,50,46)(12,27,51,47)(13,39,33,19)(14,40,34,20)(15,41,35,21)(16,42,36,22)(17,37,31,23)(18,38,32,24)(55,71,91,85)(56,72,92,86)(57,67,93,87)(58,68,94,88)(59,69,95,89)(60,70,96,90), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,95,52,59)(8,58,53,94)(9,93,54,57)(10,56,49,92)(11,91,50,55)(12,60,51,96)(13,16,33,36)(14,35,34,15)(17,18,31,32)(19,66,39,80)(20,79,40,65)(21,64,41,84)(22,83,42,63)(23,62,37,82)(24,81,38,61)(25,28,45,48)(26,47,46,27)(29,30,43,44)(67,68,87,88)(69,72,89,86)(70,85,90,71)>;
G:=Group( (1,30,15,68)(2,25,16,69)(3,26,17,70)(4,27,18,71)(5,28,13,72)(6,29,14,67)(7,19,56,63)(8,20,57,64)(9,21,58,65)(10,22,59,66)(11,23,60,61)(12,24,55,62)(31,90,75,46)(32,85,76,47)(33,86,77,48)(34,87,78,43)(35,88,73,44)(36,89,74,45)(37,96,81,50)(38,91,82,51)(39,92,83,52)(40,93,84,53)(41,94,79,54)(42,95,80,49), (1,79,73,65)(2,80,74,66)(3,81,75,61)(4,82,76,62)(5,83,77,63)(6,84,78,64)(7,28,52,48)(8,29,53,43)(9,30,54,44)(10,25,49,45)(11,26,50,46)(12,27,51,47)(13,39,33,19)(14,40,34,20)(15,41,35,21)(16,42,36,22)(17,37,31,23)(18,38,32,24)(55,71,91,85)(56,72,92,86)(57,67,93,87)(58,68,94,88)(59,69,95,89)(60,70,96,90), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,95,52,59)(8,58,53,94)(9,93,54,57)(10,56,49,92)(11,91,50,55)(12,60,51,96)(13,16,33,36)(14,35,34,15)(17,18,31,32)(19,66,39,80)(20,79,40,65)(21,64,41,84)(22,83,42,63)(23,62,37,82)(24,81,38,61)(25,28,45,48)(26,47,46,27)(29,30,43,44)(67,68,87,88)(69,72,89,86)(70,85,90,71) );
G=PermutationGroup([[(1,30,15,68),(2,25,16,69),(3,26,17,70),(4,27,18,71),(5,28,13,72),(6,29,14,67),(7,19,56,63),(8,20,57,64),(9,21,58,65),(10,22,59,66),(11,23,60,61),(12,24,55,62),(31,90,75,46),(32,85,76,47),(33,86,77,48),(34,87,78,43),(35,88,73,44),(36,89,74,45),(37,96,81,50),(38,91,82,51),(39,92,83,52),(40,93,84,53),(41,94,79,54),(42,95,80,49)], [(1,79,73,65),(2,80,74,66),(3,81,75,61),(4,82,76,62),(5,83,77,63),(6,84,78,64),(7,28,52,48),(8,29,53,43),(9,30,54,44),(10,25,49,45),(11,26,50,46),(12,27,51,47),(13,39,33,19),(14,40,34,20),(15,41,35,21),(16,42,36,22),(17,37,31,23),(18,38,32,24),(55,71,91,85),(56,72,92,86),(57,67,93,87),(58,68,94,88),(59,69,95,89),(60,70,96,90)], [(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)], [(1,6,73,78),(2,77,74,5),(3,4,75,76),(7,95,52,59),(8,58,53,94),(9,93,54,57),(10,56,49,92),(11,91,50,55),(12,60,51,96),(13,16,33,36),(14,35,34,15),(17,18,31,32),(19,66,39,80),(20,79,40,65),(21,64,41,84),(22,83,42,63),(23,62,37,82),(24,81,38,61),(25,28,45,48),(26,47,46,27),(29,30,43,44),(67,68,87,88),(69,72,89,86),(70,85,90,71)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | C4○D12 |
| kernel | C42.277D6 | C4×Dic6 | C12.6Q8 | C42⋊2S3 | C4×D12 | C42⋊7S3 | C42⋊3S3 | C12.48D4 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | C2×C4×C12 | C2×C42 | C42 | C22×C4 | C12 | C2×C6 | C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 8 | 4 | 16 | 8 |
Matrix representation of C42.277D6 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
| 1 | 11 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 2 | 11 |
| 0 | 0 | 2 | 4 |
| 12 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 11 | 2 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,2,9,0,0,4,11],[1,0,0,0,11,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,2,2,0,0,11,4],[12,12,0,0,0,1,0,0,0,0,11,11,0,0,9,2] >;
C42.277D6 in GAP, Magma, Sage, TeX
C_4^2._{277}D_6 % in TeX
G:=Group("C4^2.277D6"); // GroupNames label
G:=SmallGroup(192,1038);
// by ID
G=gap.SmallGroup(192,1038);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations