metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C3×D4).8D4, (D4×Dic3)⋊7C2, (C2×SD16)⋊8S3, (C2×C8).145D6, (C2×Q8).75D6, Dic3⋊C8⋊34C2, (C6×SD16)⋊19C2, (C2×D4).145D6, C6.60(C4○D8), C2.D24⋊34C2, C12.172(C2×D4), C3⋊7(D4.2D4), D4.3(C3⋊D4), C12.99(C4○D4), Q8⋊2Dic3⋊27C2, C12.23D4⋊3C2, C2.27(Q8⋊3D6), C6.77(C8⋊C22), (C2×Dic3).68D4, (C6×D4).91C22, C22.262(S3×D4), (C6×Q8).72C22, C4.11(D4⋊2S3), C6.114(C4⋊D4), (C2×C12).442C23, (C2×C24).292C22, C2.26(Q8.7D6), (C2×D12).118C22, C4⋊Dic3.172C22, (C4×Dic3).49C22, C2.26(C23.14D6), (C2×D4⋊S3).8C2, C4.40(C2×C3⋊D4), (C2×C6).354(C2×D4), (C2×C3⋊C8).154C22, (C2×C4).531(C22×S3), SmallGroup(192,724)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×Dic3)⋊C2
G = < a,b,c,d,e | a4=b2=c6=e2=1, d2=c3, bab=eae=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe=ab, dcd-1=ece=c-1, ede=a2c3d >
Subgroups: 392 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×7], S3, C6 [×3], C6 [×2], C8 [×2], C2×C4, C2×C4 [×6], D4 [×2], D4 [×3], Q8 [×2], C23 [×2], Dic3 [×3], C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, D12 [×2], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4 [×2], D4⋊S3 [×2], C6.D4, C2×C24, C3×SD16 [×2], C2×D12, C22×Dic3, C6×D4, C6×Q8, D4.2D4, Dic3⋊C8, C2.D24, Q8⋊2Dic3, C2×D4⋊S3, D4×Dic3, C12.23D4, C6×SD16, (D4×Dic3)⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8⋊C22, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.2D4, Q8⋊3D6, Q8.7D6, C23.14D6, (D4×Dic3)⋊C2
►On 96 points
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 87 94 82)(8 88 95 83)(9 89 96 84)(10 90 91 79)(11 85 92 80)(12 86 93 81)(31 43 42 54)(32 44 37 49)(33 45 38 50)(34 46 39 51)(35 47 40 52)(36 48 41 53)(55 67 66 78)(56 68 61 73)(57 69 62 74)(58 70 63 75)(59 71 64 76)(60 72 65 77)
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 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 36 4 33)(2 35 5 32)(3 34 6 31)(7 71 10 68)(8 70 11 67)(9 69 12 72)(13 39 16 42)(14 38 17 41)(15 37 18 40)(19 52 22 49)(20 51 23 54)(21 50 24 53)(25 46 28 43)(26 45 29 48)(27 44 30 47)(55 83 58 80)(56 82 59 79)(57 81 60 84)(61 87 64 90)(62 86 65 89)(63 85 66 88)(73 94 76 91)(74 93 77 96)(75 92 78 95)
(2 6)(3 5)(7 83)(8 82)(9 81)(10 80)(11 79)(12 84)(13 15)(16 18)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)(43 44)(45 48)(46 47)(49 54)(50 53)(51 52)(55 76)(56 75)(57 74)(58 73)(59 78)(60 77)(61 70)(62 69)(63 68)(64 67)(65 72)(66 71)(85 91)(86 96)(87 95)(88 94)(89 93)(90 92)
G:=sub<Sym(96)| (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,87,94,82)(8,88,95,83)(9,89,96,84)(10,90,91,79)(11,85,92,80)(12,86,93,81)(31,43,42,54)(32,44,37,49)(33,45,38,50)(34,46,39,51)(35,47,40,52)(36,48,41,53)(55,67,66,78)(56,68,61,73)(57,69,62,74)(58,70,63,75)(59,71,64,76)(60,72,65,77), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,36,4,33)(2,35,5,32)(3,34,6,31)(7,71,10,68)(8,70,11,67)(9,69,12,72)(13,39,16,42)(14,38,17,41)(15,37,18,40)(19,52,22,49)(20,51,23,54)(21,50,24,53)(25,46,28,43)(26,45,29,48)(27,44,30,47)(55,83,58,80)(56,82,59,79)(57,81,60,84)(61,87,64,90)(62,86,65,89)(63,85,66,88)(73,94,76,91)(74,93,77,96)(75,92,78,95), (2,6)(3,5)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,15)(16,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,44)(45,48)(46,47)(49,54)(50,53)(51,52)(55,76)(56,75)(57,74)(58,73)(59,78)(60,77)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92)>;
G:=Group( (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,87,94,82)(8,88,95,83)(9,89,96,84)(10,90,91,79)(11,85,92,80)(12,86,93,81)(31,43,42,54)(32,44,37,49)(33,45,38,50)(34,46,39,51)(35,47,40,52)(36,48,41,53)(55,67,66,78)(56,68,61,73)(57,69,62,74)(58,70,63,75)(59,71,64,76)(60,72,65,77), (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,36,4,33)(2,35,5,32)(3,34,6,31)(7,71,10,68)(8,70,11,67)(9,69,12,72)(13,39,16,42)(14,38,17,41)(15,37,18,40)(19,52,22,49)(20,51,23,54)(21,50,24,53)(25,46,28,43)(26,45,29,48)(27,44,30,47)(55,83,58,80)(56,82,59,79)(57,81,60,84)(61,87,64,90)(62,86,65,89)(63,85,66,88)(73,94,76,91)(74,93,77,96)(75,92,78,95), (2,6)(3,5)(7,83)(8,82)(9,81)(10,80)(11,79)(12,84)(13,15)(16,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)(43,44)(45,48)(46,47)(49,54)(50,53)(51,52)(55,76)(56,75)(57,74)(58,73)(59,78)(60,77)(61,70)(62,69)(63,68)(64,67)(65,72)(66,71)(85,91)(86,96)(87,95)(88,94)(89,93)(90,92) );
G=PermutationGroup([(1,29,17,24),(2,30,18,19),(3,25,13,20),(4,26,14,21),(5,27,15,22),(6,28,16,23),(7,87,94,82),(8,88,95,83),(9,89,96,84),(10,90,91,79),(11,85,92,80),(12,86,93,81),(31,43,42,54),(32,44,37,49),(33,45,38,50),(34,46,39,51),(35,47,40,52),(36,48,41,53),(55,67,66,78),(56,68,61,73),(57,69,62,74),(58,70,63,75),(59,71,64,76),(60,72,65,77)], [(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(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)], [(1,36,4,33),(2,35,5,32),(3,34,6,31),(7,71,10,68),(8,70,11,67),(9,69,12,72),(13,39,16,42),(14,38,17,41),(15,37,18,40),(19,52,22,49),(20,51,23,54),(21,50,24,53),(25,46,28,43),(26,45,29,48),(27,44,30,47),(55,83,58,80),(56,82,59,79),(57,81,60,84),(61,87,64,90),(62,86,65,89),(63,85,66,88),(73,94,76,91),(74,93,77,96),(75,92,78,95)], [(2,6),(3,5),(7,83),(8,82),(9,81),(10,80),(11,79),(12,84),(13,15),(16,18),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38),(43,44),(45,48),(46,47),(49,54),(50,53),(51,52),(55,76),(56,75),(57,74),(58,73),(59,78),(60,77),(61,70),(62,69),(63,68),(64,67),(65,72),(66,71),(85,91),(86,96),(87,95),(88,94),(89,93),(90,92)])
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 | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| 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 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
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 | D4⋊2S3 | S3×D4 | Q8⋊3D6 | Q8.7D6 |
| kernel | (D4×Dic3)⋊C2 | Dic3⋊C8 | C2.D24 | Q8⋊2Dic3 | C2×D4⋊S3 | D4×Dic3 | C12.23D4 | C6×SD16 | C2×SD16 | C2×Dic3 | C3×D4 | C2×C8 | C2×D4 | C2×Q8 | C12 | D4 | C6 | C6 | C4 | C22 | 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 (D4×Dic3)⋊C2 ►in GL4(𝔽73) generated by
| 72 | 71 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 41 | 41 | 0 | 0 |
| 16 | 32 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [72,1,0,0,71,1,0,0,0,0,1,0,0,0,0,1],[41,16,0,0,41,32,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[27,0,0,0,0,27,0,0,0,0,60,30,0,0,43,13],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
(D4×Dic3)⋊C2 in GAP, Magma, Sage, TeX
(D_4\times {\rm Dic}_3)\rtimes C_2 % in TeX
G:=Group("(D4xDic3):C2"); // GroupNames label
G:=SmallGroup(192,724);
// by ID
G=gap.SmallGroup(192,724);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1094,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^6=e^2=1,d^2=c^3,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e=a*b,d*c*d^-1=e*c*e=c^-1,e*d*e=a^2*c^3*d>;
// generators/relations