metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊5Q8, C42.79D6, C12.18SD16, C4⋊Q8⋊2S3, C4⋊C4.82D6, C4.11(S3×Q8), C12⋊C8⋊34C2, C3⋊5(D4⋊2Q8), C12.38(C2×Q8), (C4×D12).18C2, (C2×C12).155D4, C6.76(C2×SD16), C12.81(C4○D4), C6.98(C8⋊C22), C12.Q8⋊42C2, C6.D8.14C2, C6.75(C22⋊Q8), (C2×C12).402C23, (C4×C12).131C22, C4.34(Q8⋊3S3), C2.12(D6⋊3Q8), C4.10(Q8⋊2S3), C2.19(D12⋊6C22), (C2×D12).248C22, C4⋊Dic3.347C22, (C3×C4⋊Q8)⋊2C2, (C2×C6).533(C2×D4), (C2×C3⋊C8).136C22, C2.14(C2×Q8⋊2S3), (C2×C4).188(C3⋊D4), (C3×C4⋊C4).129C22, (C2×C4).499(C22×S3), C22.205(C2×C3⋊D4), SmallGroup(192,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊5Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a6b, dcd-1=c-1 >
Subgroups: 320 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, D4⋊2Q8, C12⋊C8, C12.Q8, C6.D8, C4×D12, C3×C4⋊Q8, D12⋊5Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, Q8⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, D4⋊2Q8, D12⋊6C22, C2×Q8⋊2S3, D6⋊3Q8, D12⋊5Q8
►On 96 points
Generators in S96
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 60)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 72)(46 71)(47 70)(48 69)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 96)(83 95)(84 94)
(1 89 34 75)(2 96 35 82)(3 91 36 77)(4 86 25 84)(5 93 26 79)(6 88 27 74)(7 95 28 81)(8 90 29 76)(9 85 30 83)(10 92 31 78)(11 87 32 73)(12 94 33 80)(13 62 50 37)(14 69 51 44)(15 64 52 39)(16 71 53 46)(17 66 54 41)(18 61 55 48)(19 68 56 43)(20 63 57 38)(21 70 58 45)(22 65 59 40)(23 72 60 47)(24 67 49 42)
(1 18 34 55)(2 19 35 56)(3 20 36 57)(4 21 25 58)(5 22 26 59)(6 23 27 60)(7 24 28 49)(8 13 29 50)(9 14 30 51)(10 15 31 52)(11 16 32 53)(12 17 33 54)(37 76 62 90)(38 77 63 91)(39 78 64 92)(40 79 65 93)(41 80 66 94)(42 81 67 95)(43 82 68 96)(44 83 69 85)(45 84 70 86)(46 73 71 87)(47 74 72 88)(48 75 61 89)
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,60)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,72)(46,71)(47,70)(48,69)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,96)(83,95)(84,94), (1,89,34,75)(2,96,35,82)(3,91,36,77)(4,86,25,84)(5,93,26,79)(6,88,27,74)(7,95,28,81)(8,90,29,76)(9,85,30,83)(10,92,31,78)(11,87,32,73)(12,94,33,80)(13,62,50,37)(14,69,51,44)(15,64,52,39)(16,71,53,46)(17,66,54,41)(18,61,55,48)(19,68,56,43)(20,63,57,38)(21,70,58,45)(22,65,59,40)(23,72,60,47)(24,67,49,42), (1,18,34,55)(2,19,35,56)(3,20,36,57)(4,21,25,58)(5,22,26,59)(6,23,27,60)(7,24,28,49)(8,13,29,50)(9,14,30,51)(10,15,31,52)(11,16,32,53)(12,17,33,54)(37,76,62,90)(38,77,63,91)(39,78,64,92)(40,79,65,93)(41,80,66,94)(42,81,67,95)(43,82,68,96)(44,83,69,85)(45,84,70,86)(46,73,71,87)(47,74,72,88)(48,75,61,89)>;
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,60)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,72)(46,71)(47,70)(48,69)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,96)(83,95)(84,94), (1,89,34,75)(2,96,35,82)(3,91,36,77)(4,86,25,84)(5,93,26,79)(6,88,27,74)(7,95,28,81)(8,90,29,76)(9,85,30,83)(10,92,31,78)(11,87,32,73)(12,94,33,80)(13,62,50,37)(14,69,51,44)(15,64,52,39)(16,71,53,46)(17,66,54,41)(18,61,55,48)(19,68,56,43)(20,63,57,38)(21,70,58,45)(22,65,59,40)(23,72,60,47)(24,67,49,42), (1,18,34,55)(2,19,35,56)(3,20,36,57)(4,21,25,58)(5,22,26,59)(6,23,27,60)(7,24,28,49)(8,13,29,50)(9,14,30,51)(10,15,31,52)(11,16,32,53)(12,17,33,54)(37,76,62,90)(38,77,63,91)(39,78,64,92)(40,79,65,93)(41,80,66,94)(42,81,67,95)(43,82,68,96)(44,83,69,85)(45,84,70,86)(46,73,71,87)(47,74,72,88)(48,75,61,89) );
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)], [(1,27),(2,26),(3,25),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,60),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,72),(46,71),(47,70),(48,69),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,96),(83,95),(84,94)], [(1,89,34,75),(2,96,35,82),(3,91,36,77),(4,86,25,84),(5,93,26,79),(6,88,27,74),(7,95,28,81),(8,90,29,76),(9,85,30,83),(10,92,31,78),(11,87,32,73),(12,94,33,80),(13,62,50,37),(14,69,51,44),(15,64,52,39),(16,71,53,46),(17,66,54,41),(18,61,55,48),(19,68,56,43),(20,63,57,38),(21,70,58,45),(22,65,59,40),(23,72,60,47),(24,67,49,42)], [(1,18,34,55),(2,19,35,56),(3,20,36,57),(4,21,25,58),(5,22,26,59),(6,23,27,60),(7,24,28,49),(8,13,29,50),(9,14,30,51),(10,15,31,52),(11,16,32,53),(12,17,33,54),(37,76,62,90),(38,77,63,91),(39,78,64,92),(40,79,65,93),(41,80,66,94),(42,81,67,95),(43,82,68,96),(44,83,69,85),(45,84,70,86),(46,73,71,87),(47,74,72,88),(48,75,61,89)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | SD16 | C4○D4 | C3⋊D4 | C8⋊C22 | Q8⋊2S3 | S3×Q8 | Q8⋊3S3 | D12⋊6C22 |
| kernel | D12⋊5Q8 | C12⋊C8 | C12.Q8 | C6.D8 | C4×D12 | C3×C4⋊Q8 | C4⋊Q8 | D12 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of D12⋊5Q8 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 62 | 72 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 71 |
| 0 | 0 | 0 | 0 | 61 | 11 |
| 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 | 46 | 0 |
| 0 | 0 | 0 | 0 | 5 | 27 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,62,0,0,0,0,0,72],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,62,61,0,0,0,0,71,11],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,5,0,0,0,0,0,27] >;
D12⋊5Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5Q_8 % in TeX
G:=Group("D12:5Q8"); // GroupNames label
G:=SmallGroup(192,643);
// by ID
G=gap.SmallGroup(192,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,268,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations