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, C2.7(C22×D12), C22.6(C2×D12), C42⋊7S3⋊33C2, D6⋊C4.80C22, (C22×C4).456D6, (C4×C12).315C22, (C2×C12).694C23, (C22×S3).3C23, C22.64(S3×C23), (C2×Dic3).5C23, (C2×D12).203C22, C4⋊Dic3.289C22, (C22×C6).383C23, C23.228(C22×S3), 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
Subgroups: 824 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×20], Q8 [×4], C23, C23 [×4], Dic3 [×4], C12 [×8], C12 [×2], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×12], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C2×C12 [×4], C22×S3 [×4], C22×C6, C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C4×C12 [×2], C4×C12 [×2], C2×Dic6 [×2], S3×C2×C4 [×4], C2×D12 [×6], C4○D12 [×8], C2×C3⋊D4 [×4], C22×C12, C22×C12 [×2], C22.26C24, C12⋊2Q8, C4×D12 [×4], C4⋊D12, C42⋊7S3 [×2], C12⋊7D4 [×4], C2×C4×C12, C2×C4○D12 [×2], C42.276D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×D12 [×6], C4○D12 [×4], S3×C23, C22.26C24, C22×D12, C2×C4○D12 [×2], C42.276D6
Generators and relations
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 >
►On 96 points
(1 67 19 55)(2 68 20 56)(3 69 21 57)(4 70 22 58)(5 71 23 59)(6 72 24 60)(7 77 29 66)(8 78 30 61)(9 73 25 62)(10 74 26 63)(11 75 27 64)(12 76 28 65)(13 51 88 40)(14 52 89 41)(15 53 90 42)(16 54 85 37)(17 49 86 38)(18 50 87 39)(31 91 43 79)(32 92 44 80)(33 93 45 81)(34 94 46 82)(35 95 47 83)(36 96 48 84)
(1 42 7 31)(2 37 8 32)(3 38 9 33)(4 39 10 34)(5 40 11 35)(6 41 12 36)(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 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 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 60 7 65)(2 64 8 59)(3 58 9 63)(4 62 10 57)(5 56 11 61)(6 66 12 55)(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 84 42 89)(32 88 37 83)(33 82 38 87)(34 86 39 81)(35 80 40 85)(36 90 41 79)
G:=sub<Sym(96)| (1,67,19,55)(2,68,20,56)(3,69,21,57)(4,70,22,58)(5,71,23,59)(6,72,24,60)(7,77,29,66)(8,78,30,61)(9,73,25,62)(10,74,26,63)(11,75,27,64)(12,76,28,65)(13,51,88,40)(14,52,89,41)(15,53,90,42)(16,54,85,37)(17,49,86,38)(18,50,87,39)(31,91,43,79)(32,92,44,80)(33,93,45,81)(34,94,46,82)(35,95,47,83)(36,96,48,84), (1,42,7,31)(2,37,8,32)(3,38,9,33)(4,39,10,34)(5,40,11,35)(6,41,12,36)(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)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (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), (1,60,7,65)(2,64,8,59)(3,58,9,63)(4,62,10,57)(5,56,11,61)(6,66,12,55)(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)(31,84,42,89)(32,88,37,83)(33,82,38,87)(34,86,39,81)(35,80,40,85)(36,90,41,79)>;
G:=Group( (1,67,19,55)(2,68,20,56)(3,69,21,57)(4,70,22,58)(5,71,23,59)(6,72,24,60)(7,77,29,66)(8,78,30,61)(9,73,25,62)(10,74,26,63)(11,75,27,64)(12,76,28,65)(13,51,88,40)(14,52,89,41)(15,53,90,42)(16,54,85,37)(17,49,86,38)(18,50,87,39)(31,91,43,79)(32,92,44,80)(33,93,45,81)(34,94,46,82)(35,95,47,83)(36,96,48,84), (1,42,7,31)(2,37,8,32)(3,38,9,33)(4,39,10,34)(5,40,11,35)(6,41,12,36)(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)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (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), (1,60,7,65)(2,64,8,59)(3,58,9,63)(4,62,10,57)(5,56,11,61)(6,66,12,55)(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)(31,84,42,89)(32,88,37,83)(33,82,38,87)(34,86,39,81)(35,80,40,85)(36,90,41,79) );
G=PermutationGroup([(1,67,19,55),(2,68,20,56),(3,69,21,57),(4,70,22,58),(5,71,23,59),(6,72,24,60),(7,77,29,66),(8,78,30,61),(9,73,25,62),(10,74,26,63),(11,75,27,64),(12,76,28,65),(13,51,88,40),(14,52,89,41),(15,53,90,42),(16,54,85,37),(17,49,86,38),(18,50,87,39),(31,91,43,79),(32,92,44,80),(33,93,45,81),(34,94,46,82),(35,95,47,83),(36,96,48,84)], [(1,42,7,31),(2,37,8,32),(3,38,9,33),(4,39,10,34),(5,40,11,35),(6,41,12,36),(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),(55,90,66,79),(56,85,61,80),(57,86,62,81),(58,87,63,82),(59,88,64,83),(60,89,65,84)], [(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)], [(1,60,7,65),(2,64,8,59),(3,58,9,63),(4,62,10,57),(5,56,11,61),(6,66,12,55),(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),(31,84,42,89),(32,88,37,83),(33,82,38,87),(34,86,39,81),(35,80,40,85),(36,90,41,79)])
Matrix representation ►G ⊆ 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] >;
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 |
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