metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.276D6, (C2×C4)⋊8D12, (C4×D12)⋊3C2, (C2×C12)⋊31D4, C4⋊5(C4○D12), (C2×C42)⋊12S3, C4.90(C2×D12), C12⋊11(C4○D4), C12⋊7D4⋊50C2, C4⋊D12⋊18C2, C6.5(C22×D4), C12⋊2Q8⋊38C2, C12.307(C2×D4), (C2×C6).21C24, C22.6(C2×D12), C2.7(C22×D12), C42⋊7S3⋊33C2, D6⋊C4.80C22, (C22×C4).456D6, (C2×C12).694C23, (C4×C12).315C22, (C22×S3).3C23, C22.64(S3×C23), (C2×Dic3).5C23, (C2×D12).203C22, C4⋊Dic3.289C22, C23.228(C22×S3), (C22×C6).383C23, C3⋊1(C22.26C24), (C22×C12).524C22, (C2×Dic6).224C22, (C2×C4×C12)⋊13C2, C6.8(C2×C4○D4), (C2×C4○D12)⋊2C2, C2.10(C2×C4○D12), (C2×C6).172(C2×D4), (S3×C2×C4).187C22, (C2×C4).730(C22×S3), (C2×C3⋊D4).85C22, SmallGroup(192,1036)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.276D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
Subgroups: 824 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, 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, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C22.26C24, C12⋊2Q8, C4×D12, C4⋊D12, C42⋊7S3, C12⋊7D4, C2×C4×C12, C2×C4○D12, C42.276D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C22×D4, C2×C4○D4, C2×D12, C4○D12, S3×C23, C22.26C24, C22×D12, C2×C4○D12, C42.276D6
►On 96 points
(1 67 19 62)(2 68 20 63)(3 69 21 64)(4 70 22 65)(5 71 23 66)(6 72 24 61)(7 78 30 55)(8 73 25 56)(9 74 26 57)(10 75 27 58)(11 76 28 59)(12 77 29 60)(13 51 82 34)(14 52 83 35)(15 53 84 36)(16 54 79 31)(17 49 80 32)(18 50 81 33)(37 96 48 85)(38 91 43 86)(39 92 44 87)(40 93 45 88)(41 94 46 89)(42 95 47 90)
(1 36 12 38)(2 31 7 39)(3 32 8 40)(4 33 9 41)(5 34 10 42)(6 35 11 37)(13 75 95 71)(14 76 96 72)(15 77 91 67)(16 78 92 68)(17 73 93 69)(18 74 94 70)(19 53 29 43)(20 54 30 44)(21 49 25 45)(22 50 26 46)(23 51 27 47)(24 52 28 48)(55 87 63 79)(56 88 64 80)(57 89 65 81)(58 90 66 82)(59 85 61 83)(60 86 62 84)
(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 61 12 59)(2 58 7 66)(3 65 8 57)(4 56 9 64)(5 63 10 55)(6 60 11 62)(13 54 95 44)(14 43 96 53)(15 52 91 48)(16 47 92 51)(17 50 93 46)(18 45 94 49)(19 72 29 76)(20 75 30 71)(21 70 25 74)(22 73 26 69)(23 68 27 78)(24 77 28 67)(31 90 39 82)(32 81 40 89)(33 88 41 80)(34 79 42 87)(35 86 37 84)(36 83 38 85)
G:=sub<Sym(96)| (1,67,19,62)(2,68,20,63)(3,69,21,64)(4,70,22,65)(5,71,23,66)(6,72,24,61)(7,78,30,55)(8,73,25,56)(9,74,26,57)(10,75,27,58)(11,76,28,59)(12,77,29,60)(13,51,82,34)(14,52,83,35)(15,53,84,36)(16,54,79,31)(17,49,80,32)(18,50,81,33)(37,96,48,85)(38,91,43,86)(39,92,44,87)(40,93,45,88)(41,94,46,89)(42,95,47,90), (1,36,12,38)(2,31,7,39)(3,32,8,40)(4,33,9,41)(5,34,10,42)(6,35,11,37)(13,75,95,71)(14,76,96,72)(15,77,91,67)(16,78,92,68)(17,73,93,69)(18,74,94,70)(19,53,29,43)(20,54,30,44)(21,49,25,45)(22,50,26,46)(23,51,27,47)(24,52,28,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84), (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), (1,61,12,59)(2,58,7,66)(3,65,8,57)(4,56,9,64)(5,63,10,55)(6,60,11,62)(13,54,95,44)(14,43,96,53)(15,52,91,48)(16,47,92,51)(17,50,93,46)(18,45,94,49)(19,72,29,76)(20,75,30,71)(21,70,25,74)(22,73,26,69)(23,68,27,78)(24,77,28,67)(31,90,39,82)(32,81,40,89)(33,88,41,80)(34,79,42,87)(35,86,37,84)(36,83,38,85)>;
G:=Group( (1,67,19,62)(2,68,20,63)(3,69,21,64)(4,70,22,65)(5,71,23,66)(6,72,24,61)(7,78,30,55)(8,73,25,56)(9,74,26,57)(10,75,27,58)(11,76,28,59)(12,77,29,60)(13,51,82,34)(14,52,83,35)(15,53,84,36)(16,54,79,31)(17,49,80,32)(18,50,81,33)(37,96,48,85)(38,91,43,86)(39,92,44,87)(40,93,45,88)(41,94,46,89)(42,95,47,90), (1,36,12,38)(2,31,7,39)(3,32,8,40)(4,33,9,41)(5,34,10,42)(6,35,11,37)(13,75,95,71)(14,76,96,72)(15,77,91,67)(16,78,92,68)(17,73,93,69)(18,74,94,70)(19,53,29,43)(20,54,30,44)(21,49,25,45)(22,50,26,46)(23,51,27,47)(24,52,28,48)(55,87,63,79)(56,88,64,80)(57,89,65,81)(58,90,66,82)(59,85,61,83)(60,86,62,84), (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), (1,61,12,59)(2,58,7,66)(3,65,8,57)(4,56,9,64)(5,63,10,55)(6,60,11,62)(13,54,95,44)(14,43,96,53)(15,52,91,48)(16,47,92,51)(17,50,93,46)(18,45,94,49)(19,72,29,76)(20,75,30,71)(21,70,25,74)(22,73,26,69)(23,68,27,78)(24,77,28,67)(31,90,39,82)(32,81,40,89)(33,88,41,80)(34,79,42,87)(35,86,37,84)(36,83,38,85) );
G=PermutationGroup([[(1,67,19,62),(2,68,20,63),(3,69,21,64),(4,70,22,65),(5,71,23,66),(6,72,24,61),(7,78,30,55),(8,73,25,56),(9,74,26,57),(10,75,27,58),(11,76,28,59),(12,77,29,60),(13,51,82,34),(14,52,83,35),(15,53,84,36),(16,54,79,31),(17,49,80,32),(18,50,81,33),(37,96,48,85),(38,91,43,86),(39,92,44,87),(40,93,45,88),(41,94,46,89),(42,95,47,90)], [(1,36,12,38),(2,31,7,39),(3,32,8,40),(4,33,9,41),(5,34,10,42),(6,35,11,37),(13,75,95,71),(14,76,96,72),(15,77,91,67),(16,78,92,68),(17,73,93,69),(18,74,94,70),(19,53,29,43),(20,54,30,44),(21,49,25,45),(22,50,26,46),(23,51,27,47),(24,52,28,48),(55,87,63,79),(56,88,64,80),(57,89,65,81),(58,90,66,82),(59,85,61,83),(60,86,62,84)], [(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)], [(1,61,12,59),(2,58,7,66),(3,65,8,57),(4,56,9,64),(5,63,10,55),(6,60,11,62),(13,54,95,44),(14,43,96,53),(15,52,91,48),(16,47,92,51),(17,50,93,46),(18,45,94,49),(19,72,29,76),(20,75,30,71),(21,70,25,74),(22,73,26,69),(23,68,27,78),(24,77,28,67),(31,90,39,82),(32,81,40,89),(33,88,41,80),(34,79,42,87),(35,86,37,84),(36,83,38,85)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C4○D12 |
| kernel | C42.276D6 | C12⋊2Q8 | C4×D12 | C4⋊D12 | C42⋊7S3 | C12⋊7D4 | C2×C4×C12 | C2×C4○D12 | C2×C42 | C2×C12 | C42 | C22×C4 | C12 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 2 | 4 | 1 | 2 | 1 | 4 | 4 | 3 | 8 | 8 | 16 |
Matrix representation of C42.276D6 ►in GL4(𝔽13) generated by
| 0 | 8 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 11 |
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 9 | 2 |
G:=sub<GL(4,GF(13))| [0,8,0,0,8,0,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,9,11,0,0,2,11],[5,0,0,0,0,8,0,0,0,0,11,9,0,0,11,2] >;
C42.276D6 in GAP, Magma, Sage, TeX
C_4^2._{276}D_6 % in TeX
G:=Group("C4^2.276D6"); // GroupNames label
G:=SmallGroup(192,1036);
// by ID
G=gap.SmallGroup(192,1036);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,6278]);
// 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^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations