metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.87D6, C6.462- 1+4, C4⋊C4.308D6, (C4×Dic6)⋊5C2, (C2×Dic6)⋊18C4, (C2×C6).60C24, C6.15(C23×C4), C2.1(Q8○D12), (C4×C12).20C22, C22⋊C4.123D6, Dic6.31(C2×C4), (C22×C4).201D6, Dic6⋊C4⋊11C2, C12.119(C22×C4), (C2×C12).581C23, C42⋊C2.10S3, C22.25(S3×C23), Dic3.6(C22×C4), C4⋊Dic3.396C22, C23.159(C22×S3), (C22×C6).130C23, (C22×Dic6).17C2, (C4×Dic3).67C22, C23.16D6.4C2, Dic3⋊C4.130C22, (C22×C12).221C22, C3⋊1(C23.32C23), (C2×Dic6).283C22, (C2×Dic3).192C23, C23.26D6.21C2, C6.D4.90C22, (C22×Dic3).84C22, C4.57(S3×C2×C4), (C2×C4).57(C4×S3), C2.17(S3×C22×C4), C22.25(S3×C2×C4), (C2×C12).129(C2×C4), (C2×C6).19(C22×C4), (C3×C4⋊C4).301C22, (C2×C4).268(C22×S3), (C2×Dic3).36(C2×C4), (C3×C42⋊C2).11C2, (C3×C22⋊C4).133C22, SmallGroup(192,1075)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.87D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c-1 >
Subgroups: 488 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, C42⋊C2, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C23.32C23, C4×Dic6, C23.16D6, Dic6⋊C4, C23.26D6, C3×C42⋊C2, C22×Dic6, C42.87D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, 2- 1+4, S3×C2×C4, S3×C23, C23.32C23, S3×C22×C4, Q8○D12, C42.87D6
►On 96 points
(1 28 4 25)(2 29 5 26)(3 30 6 27)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 18 47 42)(14 16 48 40)(15 17 46 41)(19 39 24 43)(20 37 22 44)(21 38 23 45)(49 64 52 61)(50 65 53 62)(51 66 54 63)(55 93 58 96)(56 94 59 91)(57 95 60 92)(67 76 70 73)(68 77 71 74)(69 78 72 75)(79 85 82 88)(80 86 83 89)(81 87 84 90)
(1 49 7 93)(2 53 8 91)(3 51 9 95)(4 52 10 96)(5 50 11 94)(6 54 12 92)(13 68 45 86)(14 72 43 90)(15 70 44 88)(16 75 19 81)(17 73 20 79)(18 77 21 83)(22 82 41 76)(23 80 42 74)(24 84 40 78)(25 61 31 55)(26 65 32 59)(27 63 33 57)(28 64 34 58)(29 62 35 56)(30 66 36 60)(37 85 46 67)(38 89 47 71)(39 87 48 69)
(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 22 4 20)(2 24 5 19)(3 23 6 21)(7 41 10 17)(8 40 11 16)(9 42 12 18)(13 36 47 33)(14 35 48 32)(15 34 46 31)(25 44 28 37)(26 43 29 39)(27 45 30 38)(49 79 52 82)(50 84 53 81)(51 83 54 80)(55 67 58 70)(56 72 59 69)(57 71 60 68)(61 85 64 88)(62 90 65 87)(63 89 66 86)(73 96 76 93)(74 95 77 92)(75 94 78 91)
G:=sub<Sym(96)| (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,18,47,42)(14,16,48,40)(15,17,46,41)(19,39,24,43)(20,37,22,44)(21,38,23,45)(49,64,52,61)(50,65,53,62)(51,66,54,63)(55,93,58,96)(56,94,59,91)(57,95,60,92)(67,76,70,73)(68,77,71,74)(69,78,72,75)(79,85,82,88)(80,86,83,89)(81,87,84,90), (1,49,7,93)(2,53,8,91)(3,51,9,95)(4,52,10,96)(5,50,11,94)(6,54,12,92)(13,68,45,86)(14,72,43,90)(15,70,44,88)(16,75,19,81)(17,73,20,79)(18,77,21,83)(22,82,41,76)(23,80,42,74)(24,84,40,78)(25,61,31,55)(26,65,32,59)(27,63,33,57)(28,64,34,58)(29,62,35,56)(30,66,36,60)(37,85,46,67)(38,89,47,71)(39,87,48,69), (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), (1,22,4,20)(2,24,5,19)(3,23,6,21)(7,41,10,17)(8,40,11,16)(9,42,12,18)(13,36,47,33)(14,35,48,32)(15,34,46,31)(25,44,28,37)(26,43,29,39)(27,45,30,38)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,67,58,70)(56,72,59,69)(57,71,60,68)(61,85,64,88)(62,90,65,87)(63,89,66,86)(73,96,76,93)(74,95,77,92)(75,94,78,91)>;
G:=Group( (1,28,4,25)(2,29,5,26)(3,30,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,18,47,42)(14,16,48,40)(15,17,46,41)(19,39,24,43)(20,37,22,44)(21,38,23,45)(49,64,52,61)(50,65,53,62)(51,66,54,63)(55,93,58,96)(56,94,59,91)(57,95,60,92)(67,76,70,73)(68,77,71,74)(69,78,72,75)(79,85,82,88)(80,86,83,89)(81,87,84,90), (1,49,7,93)(2,53,8,91)(3,51,9,95)(4,52,10,96)(5,50,11,94)(6,54,12,92)(13,68,45,86)(14,72,43,90)(15,70,44,88)(16,75,19,81)(17,73,20,79)(18,77,21,83)(22,82,41,76)(23,80,42,74)(24,84,40,78)(25,61,31,55)(26,65,32,59)(27,63,33,57)(28,64,34,58)(29,62,35,56)(30,66,36,60)(37,85,46,67)(38,89,47,71)(39,87,48,69), (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), (1,22,4,20)(2,24,5,19)(3,23,6,21)(7,41,10,17)(8,40,11,16)(9,42,12,18)(13,36,47,33)(14,35,48,32)(15,34,46,31)(25,44,28,37)(26,43,29,39)(27,45,30,38)(49,79,52,82)(50,84,53,81)(51,83,54,80)(55,67,58,70)(56,72,59,69)(57,71,60,68)(61,85,64,88)(62,90,65,87)(63,89,66,86)(73,96,76,93)(74,95,77,92)(75,94,78,91) );
G=PermutationGroup([[(1,28,4,25),(2,29,5,26),(3,30,6,27),(7,34,10,31),(8,35,11,32),(9,36,12,33),(13,18,47,42),(14,16,48,40),(15,17,46,41),(19,39,24,43),(20,37,22,44),(21,38,23,45),(49,64,52,61),(50,65,53,62),(51,66,54,63),(55,93,58,96),(56,94,59,91),(57,95,60,92),(67,76,70,73),(68,77,71,74),(69,78,72,75),(79,85,82,88),(80,86,83,89),(81,87,84,90)], [(1,49,7,93),(2,53,8,91),(3,51,9,95),(4,52,10,96),(5,50,11,94),(6,54,12,92),(13,68,45,86),(14,72,43,90),(15,70,44,88),(16,75,19,81),(17,73,20,79),(18,77,21,83),(22,82,41,76),(23,80,42,74),(24,84,40,78),(25,61,31,55),(26,65,32,59),(27,63,33,57),(28,64,34,58),(29,62,35,56),(30,66,36,60),(37,85,46,67),(38,89,47,71),(39,87,48,69)], [(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)], [(1,22,4,20),(2,24,5,19),(3,23,6,21),(7,41,10,17),(8,40,11,16),(9,42,12,18),(13,36,47,33),(14,35,48,32),(15,34,46,31),(25,44,28,37),(26,43,29,39),(27,45,30,38),(49,79,52,82),(50,84,53,81),(51,83,54,80),(55,67,58,70),(56,72,59,69),(57,71,60,68),(61,85,64,88),(62,90,65,87),(63,89,66,86),(73,96,76,93),(74,95,77,92),(75,94,78,91)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4L | 4M | ··· | 4AB | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4×S3 | 2- 1+4 | Q8○D12 |
| kernel | C42.87D6 | C4×Dic6 | C23.16D6 | Dic6⋊C4 | C23.26D6 | C3×C42⋊C2 | C22×Dic6 | C2×Dic6 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
Matrix representation of C42.87D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 12 | 8 | 3 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 12 | 12 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 6 | 6 | 6 | 12 |
| 0 | 0 | 9 | 0 | 10 | 0 |
| 0 | 0 | 4 | 11 | 7 | 7 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,12,1,0,0,0,12,0,8,0,0,1,8,0,0,0,0,0,3,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,8,0,0,0,1,0,12,0,0,0,0,0,8,0,0,0,0,0,3,5],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,12,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[3,6,0,0,0,0,3,10,0,0,0,0,0,0,3,6,9,4,0,0,0,6,0,11,0,0,9,6,10,7,0,0,0,12,0,7] >;
C42.87D6 in GAP, Magma, Sage, TeX
C_4^2._{87}D_6 % in TeX
G:=Group("C4^2.87D6"); // GroupNames label
G:=SmallGroup(192,1075);
// by ID
G=gap.SmallGroup(192,1075);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations