metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.23D4, C42.63D6, C4.50(S3×D4), (C4×D12)⋊21C2, C12⋊C8⋊29C2, (C2×D4).47D6, C12.24(C2×D4), C4.4D4⋊1S3, (C2×Q8).61D6, (C2×C12).271D4, C3⋊5(D4.2D4), C12.68(C4○D4), C6.105(C4○D8), D4⋊Dic3⋊19C2, Q8⋊2Dic3⋊22C2, C4.2(D4⋊2S3), (C6×D4).63C22, (C6×Q8).55C22, C2.18(D4⋊D6), C2.11(D6⋊3D4), C6.102(C4⋊D4), C6.119(C8⋊C22), (C2×C12).375C23, (C4×C12).106C22, C2.24(Q8.13D6), (C2×D12).244C22, C4⋊Dic3.341C22, (C2×D4⋊S3).6C2, (C2×C6).506(C2×D4), (C3×C4.4D4)⋊1C2, (C2×Q8⋊2S3)⋊12C2, (C2×C4).61(C3⋊D4), (C2×C3⋊C8).121C22, (C2×C4).475(C22×S3), C22.181(C2×C3⋊D4), SmallGroup(192,616)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.23D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >
Subgroups: 384 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, Q8⋊2S3, C4×C12, C3×C22⋊C4, S3×C2×C4, C2×D12, C6×D4, C6×Q8, D4.2D4, C12⋊C8, D4⋊Dic3, Q8⋊2Dic3, C4×D12, C2×D4⋊S3, C2×Q8⋊2S3, C3×C4.4D4, D12.23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8⋊C22, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.2D4, D6⋊3D4, D4⋊D6, Q8.13D6, D12.23D4
►On 96 points
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 84)(10 83)(11 82)(12 81)(13 26)(14 25)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 65)(50 64)(51 63)(52 62)(53 61)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 64 81 51)(2 65 82 52)(3 66 83 53)(4 67 84 54)(5 68 73 55)(6 69 74 56)(7 70 75 57)(8 71 76 58)(9 72 77 59)(10 61 78 60)(11 62 79 49)(12 63 80 50)(13 42 30 95)(14 43 31 96)(15 44 32 85)(16 45 33 86)(17 46 34 87)(18 47 35 88)(19 48 36 89)(20 37 25 90)(21 38 26 91)(22 39 27 92)(23 40 28 93)(24 41 29 94)
(1 95 7 89)(2 90 8 96)(3 85 9 91)(4 92 10 86)(5 87 11 93)(6 94 12 88)(13 51 19 57)(14 58 20 52)(15 53 21 59)(16 60 22 54)(17 55 23 49)(18 50 24 56)(25 65 31 71)(26 72 32 66)(27 67 33 61)(28 62 34 68)(29 69 35 63)(30 64 36 70)(37 76 43 82)(38 83 44 77)(39 78 45 84)(40 73 46 79)(41 80 47 74)(42 75 48 81)
G:=sub<Sym(96)| (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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,84)(10,83)(11,82)(12,81)(13,26)(14,25)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66), (1,64,81,51)(2,65,82,52)(3,66,83,53)(4,67,84,54)(5,68,73,55)(6,69,74,56)(7,70,75,57)(8,71,76,58)(9,72,77,59)(10,61,78,60)(11,62,79,49)(12,63,80,50)(13,42,30,95)(14,43,31,96)(15,44,32,85)(16,45,33,86)(17,46,34,87)(18,47,35,88)(19,48,36,89)(20,37,25,90)(21,38,26,91)(22,39,27,92)(23,40,28,93)(24,41,29,94), (1,95,7,89)(2,90,8,96)(3,85,9,91)(4,92,10,86)(5,87,11,93)(6,94,12,88)(13,51,19,57)(14,58,20,52)(15,53,21,59)(16,60,22,54)(17,55,23,49)(18,50,24,56)(25,65,31,71)(26,72,32,66)(27,67,33,61)(28,62,34,68)(29,69,35,63)(30,64,36,70)(37,76,43,82)(38,83,44,77)(39,78,45,84)(40,73,46,79)(41,80,47,74)(42,75,48,81)>;
G:=Group( (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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,84)(10,83)(11,82)(12,81)(13,26)(14,25)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,65)(50,64)(51,63)(52,62)(53,61)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66), (1,64,81,51)(2,65,82,52)(3,66,83,53)(4,67,84,54)(5,68,73,55)(6,69,74,56)(7,70,75,57)(8,71,76,58)(9,72,77,59)(10,61,78,60)(11,62,79,49)(12,63,80,50)(13,42,30,95)(14,43,31,96)(15,44,32,85)(16,45,33,86)(17,46,34,87)(18,47,35,88)(19,48,36,89)(20,37,25,90)(21,38,26,91)(22,39,27,92)(23,40,28,93)(24,41,29,94), (1,95,7,89)(2,90,8,96)(3,85,9,91)(4,92,10,86)(5,87,11,93)(6,94,12,88)(13,51,19,57)(14,58,20,52)(15,53,21,59)(16,60,22,54)(17,55,23,49)(18,50,24,56)(25,65,31,71)(26,72,32,66)(27,67,33,61)(28,62,34,68)(29,69,35,63)(30,64,36,70)(37,76,43,82)(38,83,44,77)(39,78,45,84)(40,73,46,79)(41,80,47,74)(42,75,48,81) );
G=PermutationGroup([[(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)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,84),(10,83),(11,82),(12,81),(13,26),(14,25),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,65),(50,64),(51,63),(52,62),(53,61),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66)], [(1,64,81,51),(2,65,82,52),(3,66,83,53),(4,67,84,54),(5,68,73,55),(6,69,74,56),(7,70,75,57),(8,71,76,58),(9,72,77,59),(10,61,78,60),(11,62,79,49),(12,63,80,50),(13,42,30,95),(14,43,31,96),(15,44,32,85),(16,45,33,86),(17,46,34,87),(18,47,35,88),(19,48,36,89),(20,37,25,90),(21,38,26,91),(22,39,27,92),(23,40,28,93),(24,41,29,94)], [(1,95,7,89),(2,90,8,96),(3,85,9,91),(4,92,10,86),(5,87,11,93),(6,94,12,88),(13,51,19,57),(14,58,20,52),(15,53,21,59),(16,60,22,54),(17,55,23,49),(18,50,24,56),(25,65,31,71),(26,72,32,66),(27,67,33,61),(28,62,34,68),(29,69,35,63),(30,64,36,70),(37,76,43,82),(38,83,44,77),(39,78,45,84),(40,73,46,79),(41,80,47,74),(42,75,48,81)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | C8⋊C22 | S3×D4 | D4⋊2S3 | D4⋊D6 | Q8.13D6 |
| kernel | D12.23D4 | C12⋊C8 | D4⋊Dic3 | Q8⋊2Dic3 | C4×D12 | C2×D4⋊S3 | C2×Q8⋊2S3 | C3×C4.4D4 | C4.4D4 | D12 | C2×C12 | C42 | C2×D4 | C2×Q8 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.23D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 0 | 0 | 6 | 61 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,6,0,0,0,0,61,61] >;
D12.23D4 in GAP, Magma, Sage, TeX
D_{12}._{23}D_4 % in TeX
G:=Group("D12.23D4"); // GroupNames label
G:=SmallGroup(192,616);
// by ID
G=gap.SmallGroup(192,616);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations