direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.2D4, C4⋊C8⋊5C6, (C4×D4)⋊4C6, (C6×D8).9C2, D4.2(C3×D4), (C2×D8).2C6, C4.35(C6×D4), (D4×C12)⋊33C2, Q8⋊C4⋊6C6, (C3×D4).27D4, C4.4D4⋊3C6, D4⋊C4⋊11C6, (C2×SD16)⋊12C6, (C6×SD16)⋊29C2, C12.396(C2×D4), (C2×C12).325D4, C42.18(C2×C6), C22.87(C6×D4), C6.121(C4○D8), C12.345(C4○D4), C6.146(C4⋊D4), C6.136(C8⋊C22), (C2×C24).301C22, (C2×C12).922C23, (C4×C12).260C22, (C6×D4).187C22, (C6×Q8).161C22, (C3×C4⋊C8)⋊15C2, C2.8(C3×C4○D8), C4⋊C4.55(C2×C6), (C2×C8).38(C2×C6), C4.44(C3×C4○D4), (C2×C4).30(C3×D4), (C2×Q8).7(C2×C6), (C2×D4).57(C2×C6), (C2×C6).643(C2×D4), C2.15(C3×C4⋊D4), C2.11(C3×C8⋊C22), (C3×D4⋊C4)⋊35C2, (C3×Q8⋊C4)⋊17C2, (C3×C4.4D4)⋊23C2, (C2×C4).97(C22×C6), (C3×C4⋊C4).376C22, SmallGroup(192,896)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.2D4
G = < a,b,c,d,e | a3=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >
Subgroups: 250 in 124 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C3×SD16, C22×C12, C6×D4, C6×Q8, D4.2D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C4⋊C8, D4×C12, C3×C4.4D4, C6×D8, C6×SD16, C3×D4.2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C4○D8, C8⋊C22, C6×D4, C3×C4○D4, D4.2D4, C3×C4⋊D4, C3×C4○D8, C3×C8⋊C22, C3×D4.2D4
►On 96 points
Generators in S96
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 32)(8 31)(9 34)(10 33)(11 36)(12 35)(13 38)(14 37)(15 40)(16 39)(17 42)(18 41)(19 44)(20 43)(21 46)(22 45)(23 48)(24 47)(49 74)(50 73)(51 76)(52 75)(53 78)(54 77)(55 80)(56 79)(57 82)(58 81)(59 84)(60 83)(61 86)(62 85)(63 88)(64 87)(65 90)(66 89)(67 92)(68 91)(69 94)(70 93)(71 96)(72 95)
(1 53 5 51)(2 54 6 52)(3 55 7 49)(4 56 8 50)(9 63 15 57)(10 64 16 58)(11 61 13 59)(12 62 14 60)(17 71 23 65)(18 72 24 66)(19 69 21 67)(20 70 22 68)(25 79 31 73)(26 80 32 74)(27 77 29 75)(28 78 30 76)(33 87 39 81)(34 88 40 82)(35 85 37 83)(36 86 38 84)(41 95 47 89)(42 96 48 90)(43 93 45 91)(44 94 46 92)
(1 49 3 51)(2 52 4 50)(5 55 7 53)(6 54 8 56)(9 59 11 57)(10 58 12 60)(13 63 15 61)(14 62 16 64)(17 67 19 65)(18 66 20 68)(21 71 23 69)(22 70 24 72)(25 76 27 74)(26 75 28 73)(29 80 31 78)(30 79 32 77)(33 84 35 82)(34 83 36 81)(37 88 39 86)(38 87 40 85)(41 92 43 90)(42 91 44 89)(45 96 47 94)(46 95 48 93)
G:=sub<Sym(96)| (1,19,11)(2,20,12)(3,17,9)(4,18,10)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,34)(10,33)(11,36)(12,35)(13,38)(14,37)(15,40)(16,39)(17,42)(18,41)(19,44)(20,43)(21,46)(22,45)(23,48)(24,47)(49,74)(50,73)(51,76)(52,75)(53,78)(54,77)(55,80)(56,79)(57,82)(58,81)(59,84)(60,83)(61,86)(62,85)(63,88)(64,87)(65,90)(66,89)(67,92)(68,91)(69,94)(70,93)(71,96)(72,95), (1,53,5,51)(2,54,6,52)(3,55,7,49)(4,56,8,50)(9,63,15,57)(10,64,16,58)(11,61,13,59)(12,62,14,60)(17,71,23,65)(18,72,24,66)(19,69,21,67)(20,70,22,68)(25,79,31,73)(26,80,32,74)(27,77,29,75)(28,78,30,76)(33,87,39,81)(34,88,40,82)(35,85,37,83)(36,86,38,84)(41,95,47,89)(42,96,48,90)(43,93,45,91)(44,94,46,92), (1,49,3,51)(2,52,4,50)(5,55,7,53)(6,54,8,56)(9,59,11,57)(10,58,12,60)(13,63,15,61)(14,62,16,64)(17,67,19,65)(18,66,20,68)(21,71,23,69)(22,70,24,72)(25,76,27,74)(26,75,28,73)(29,80,31,78)(30,79,32,77)(33,84,35,82)(34,83,36,81)(37,88,39,86)(38,87,40,85)(41,92,43,90)(42,91,44,89)(45,96,47,94)(46,95,48,93)>;
G:=Group( (1,19,11)(2,20,12)(3,17,9)(4,18,10)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,32)(8,31)(9,34)(10,33)(11,36)(12,35)(13,38)(14,37)(15,40)(16,39)(17,42)(18,41)(19,44)(20,43)(21,46)(22,45)(23,48)(24,47)(49,74)(50,73)(51,76)(52,75)(53,78)(54,77)(55,80)(56,79)(57,82)(58,81)(59,84)(60,83)(61,86)(62,85)(63,88)(64,87)(65,90)(66,89)(67,92)(68,91)(69,94)(70,93)(71,96)(72,95), (1,53,5,51)(2,54,6,52)(3,55,7,49)(4,56,8,50)(9,63,15,57)(10,64,16,58)(11,61,13,59)(12,62,14,60)(17,71,23,65)(18,72,24,66)(19,69,21,67)(20,70,22,68)(25,79,31,73)(26,80,32,74)(27,77,29,75)(28,78,30,76)(33,87,39,81)(34,88,40,82)(35,85,37,83)(36,86,38,84)(41,95,47,89)(42,96,48,90)(43,93,45,91)(44,94,46,92), (1,49,3,51)(2,52,4,50)(5,55,7,53)(6,54,8,56)(9,59,11,57)(10,58,12,60)(13,63,15,61)(14,62,16,64)(17,67,19,65)(18,66,20,68)(21,71,23,69)(22,70,24,72)(25,76,27,74)(26,75,28,73)(29,80,31,78)(30,79,32,77)(33,84,35,82)(34,83,36,81)(37,88,39,86)(38,87,40,85)(41,92,43,90)(42,91,44,89)(45,96,47,94)(46,95,48,93) );
G=PermutationGroup([[(1,19,11),(2,20,12),(3,17,9),(4,18,10),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,30),(6,29),(7,32),(8,31),(9,34),(10,33),(11,36),(12,35),(13,38),(14,37),(15,40),(16,39),(17,42),(18,41),(19,44),(20,43),(21,46),(22,45),(23,48),(24,47),(49,74),(50,73),(51,76),(52,75),(53,78),(54,77),(55,80),(56,79),(57,82),(58,81),(59,84),(60,83),(61,86),(62,85),(63,88),(64,87),(65,90),(66,89),(67,92),(68,91),(69,94),(70,93),(71,96),(72,95)], [(1,53,5,51),(2,54,6,52),(3,55,7,49),(4,56,8,50),(9,63,15,57),(10,64,16,58),(11,61,13,59),(12,62,14,60),(17,71,23,65),(18,72,24,66),(19,69,21,67),(20,70,22,68),(25,79,31,73),(26,80,32,74),(27,77,29,75),(28,78,30,76),(33,87,39,81),(34,88,40,82),(35,85,37,83),(36,86,38,84),(41,95,47,89),(42,96,48,90),(43,93,45,91),(44,94,46,92)], [(1,49,3,51),(2,52,4,50),(5,55,7,53),(6,54,8,56),(9,59,11,57),(10,58,12,60),(13,63,15,61),(14,62,16,64),(17,67,19,65),(18,66,20,68),(21,71,23,69),(22,70,24,72),(25,76,27,74),(26,75,28,73),(29,80,31,78),(30,79,32,77),(33,84,35,82),(34,83,36,81),(37,88,39,86),(38,87,40,85),(41,92,43,90),(42,91,44,89),(45,96,47,94),(46,95,48,93)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12N | 12O | 12P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D4 | C3×C4○D8 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×D4.2D4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C4⋊C8 | D4×C12 | C3×C4.4D4 | C6×D8 | C6×SD16 | D4.2D4 | D4⋊C4 | Q8⋊C4 | C4⋊C8 | C4×D4 | C4.4D4 | C2×D8 | C2×SD16 | C2×C12 | C3×D4 | C12 | C2×C4 | D4 | C6 | C4 | C2 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 |
Matrix representation of C3×D4.2D4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 57 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 1 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[16,57,0,0,57,57,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,0,46,0,0,0,0,72,1,0,0,71,1],[46,0,0,0,0,27,0,0,0,0,72,0,0,0,71,1] >;
C3×D4.2D4 in GAP, Magma, Sage, TeX
C_3\times D_4._2D_4
% in TeX
G:=Group("C3xD4.2D4"); // GroupNames label
G:=SmallGroup(192,896);
// by ID
G=gap.SmallGroup(192,896);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations