metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.2D12, C12⋊10SD16, C42.52D6, (C4×D4).9S3, C4⋊C4.247D6, C4⋊3(D4.S3), (C3×D4).19D4, C12⋊C8⋊25C2, C4.15(C2×D12), C12.19(C2×D4), (C2×C12).62D4, C12⋊2Q8⋊16C2, (C2×D4).194D6, (D4×C12).10C2, C3⋊3(D4.D4), C6.53(C2×SD16), C12.54(C4○D4), C4.11(C4○D12), C6.SD16⋊31C2, C6.66(C4⋊D4), (C4×C12).90C22, C2.14(C12⋊7D4), (C2×C12).341C23, C2.8(Q8.14D6), (C6×D4).236C22, C6.109(C8.C22), (C2×Dic6).100C22, C2.7(C2×D4.S3), (C2×C6).472(C2×D4), (C2×C3⋊C8).96C22, (C2×D4.S3).5C2, (C2×C4).246(C3⋊D4), (C3×C4⋊C4).278C22, (C2×C4).441(C22×S3), C22.151(C2×C3⋊D4), SmallGroup(192,578)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.2D12
G = < a,b,c,d | a4=b2=c12=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Subgroups: 312 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×C3⋊C8, C4⋊Dic3, D4.S3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, D4.D4, C12⋊C8, C6.SD16, C12⋊2Q8, C2×D4.S3, D4×C12, D4.2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8.C22, D4.S3, C2×D12, C4○D12, C2×C3⋊D4, D4.D4, C12⋊7D4, C2×D4.S3, Q8.14D6, D4.2D12
►On 96 points
(1 60 43 17)(2 49 44 18)(3 50 45 19)(4 51 46 20)(5 52 47 21)(6 53 48 22)(7 54 37 23)(8 55 38 24)(9 56 39 13)(10 57 40 14)(11 58 41 15)(12 59 42 16)(25 80 61 89)(26 81 62 90)(27 82 63 91)(28 83 64 92)(29 84 65 93)(30 73 66 94)(31 74 67 95)(32 75 68 96)(33 76 69 85)(34 77 70 86)(35 78 71 87)(36 79 72 88)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 85)(83 86)(84 87)
(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 88 43 79)(2 87 44 78)(3 86 45 77)(4 85 46 76)(5 96 47 75)(6 95 48 74)(7 94 37 73)(8 93 38 84)(9 92 39 83)(10 91 40 82)(11 90 41 81)(12 89 42 80)(13 28 56 64)(14 27 57 63)(15 26 58 62)(16 25 59 61)(17 36 60 72)(18 35 49 71)(19 34 50 70)(20 33 51 69)(21 32 52 68)(22 31 53 67)(23 30 54 66)(24 29 55 65)
G:=sub<Sym(96)| (1,60,43,17)(2,49,44,18)(3,50,45,19)(4,51,46,20)(5,52,47,21)(6,53,48,22)(7,54,37,23)(8,55,38,24)(9,56,39,13)(10,57,40,14)(11,58,41,15)(12,59,42,16)(25,80,61,89)(26,81,62,90)(27,82,63,91)(28,83,64,92)(29,84,65,93)(30,73,66,94)(31,74,67,95)(32,75,68,96)(33,76,69,85)(34,77,70,86)(35,78,71,87)(36,79,72,88), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87), (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), (1,88,43,79)(2,87,44,78)(3,86,45,77)(4,85,46,76)(5,96,47,75)(6,95,48,74)(7,94,37,73)(8,93,38,84)(9,92,39,83)(10,91,40,82)(11,90,41,81)(12,89,42,80)(13,28,56,64)(14,27,57,63)(15,26,58,62)(16,25,59,61)(17,36,60,72)(18,35,49,71)(19,34,50,70)(20,33,51,69)(21,32,52,68)(22,31,53,67)(23,30,54,66)(24,29,55,65)>;
G:=Group( (1,60,43,17)(2,49,44,18)(3,50,45,19)(4,51,46,20)(5,52,47,21)(6,53,48,22)(7,54,37,23)(8,55,38,24)(9,56,39,13)(10,57,40,14)(11,58,41,15)(12,59,42,16)(25,80,61,89)(26,81,62,90)(27,82,63,91)(28,83,64,92)(29,84,65,93)(30,73,66,94)(31,74,67,95)(32,75,68,96)(33,76,69,85)(34,77,70,86)(35,78,71,87)(36,79,72,88), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87), (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), (1,88,43,79)(2,87,44,78)(3,86,45,77)(4,85,46,76)(5,96,47,75)(6,95,48,74)(7,94,37,73)(8,93,38,84)(9,92,39,83)(10,91,40,82)(11,90,41,81)(12,89,42,80)(13,28,56,64)(14,27,57,63)(15,26,58,62)(16,25,59,61)(17,36,60,72)(18,35,49,71)(19,34,50,70)(20,33,51,69)(21,32,52,68)(22,31,53,67)(23,30,54,66)(24,29,55,65) );
G=PermutationGroup([[(1,60,43,17),(2,49,44,18),(3,50,45,19),(4,51,46,20),(5,52,47,21),(6,53,48,22),(7,54,37,23),(8,55,38,24),(9,56,39,13),(10,57,40,14),(11,58,41,15),(12,59,42,16),(25,80,61,89),(26,81,62,90),(27,82,63,91),(28,83,64,92),(29,84,65,93),(30,73,66,94),(31,74,67,95),(32,75,68,96),(33,76,69,85),(34,77,70,86),(35,78,71,87),(36,79,72,88)], [(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,60),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,85),(83,86),(84,87)], [(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)], [(1,88,43,79),(2,87,44,78),(3,86,45,77),(4,85,46,76),(5,96,47,75),(6,95,48,74),(7,94,37,73),(8,93,38,84),(9,92,39,83),(10,91,40,82),(11,90,41,81),(12,89,42,80),(13,28,56,64),(14,27,57,63),(15,26,58,62),(16,25,59,61),(17,36,60,72),(18,35,49,71),(19,34,50,70),(20,33,51,69),(21,32,52,68),(22,31,53,67),(23,30,54,66),(24,29,55,65)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | C4○D4 | C3⋊D4 | D12 | C4○D12 | C8.C22 | D4.S3 | Q8.14D6 |
| kernel | D4.2D12 | C12⋊C8 | C6.SD16 | C12⋊2Q8 | C2×D4.S3 | D4×C12 | C4×D4 | C2×C12 | C3×D4 | C42 | C4⋊C4 | C2×D4 | C12 | C12 | C2×C4 | D4 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of D4.2D12 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 11 | 0 | 0 |
| 0 | 0 | 59 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 46 | 0 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,65,2,0,0,0,0,0,9,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[67,6,0,0,0,0,6,6,0,0,0,0,0,0,70,59,0,0,0,0,11,3,0,0,0,0,0,0,0,46,0,0,0,0,27,0] >;
D4.2D12 in GAP, Magma, Sage, TeX
D_4._2D_{12} % in TeX
G:=Group("D4.2D12"); // GroupNames label
G:=SmallGroup(192,578);
// by ID
G=gap.SmallGroup(192,578);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations