direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊2D4, C24⋊17D4, C8⋊2(C3×D4), (C2×D8)⋊7C6, C4.Q8⋊4C6, (C6×D8)⋊21C2, C4⋊D4⋊4C6, C4.61(C6×D4), D4⋊C4⋊18C6, (C2×C12).328D4, C12.468(C2×D4), (C6×M4(2))⋊7C2, (C2×M4(2))⋊2C6, (C22×C6).34D4, C22.93(C6×D4), C23.16(C3×D4), C12.266(C4○D4), C6.152(C4⋊D4), C6.138(C8⋊C22), (C2×C12).928C23, (C2×C24).333C22, (C6×D4).192C22, (C22×C12).426C22, C4⋊C4.9(C2×C6), (C2×C8).22(C2×C6), (C3×C4.Q8)⋊13C2, C4.11(C3×C4○D4), (C2×C4).33(C3×D4), (C3×C4⋊D4)⋊31C2, (C2×D4).15(C2×C6), (C2×C6).649(C2×D4), C2.21(C3×C4⋊D4), C2.13(C3×C8⋊C22), (C3×D4⋊C4)⋊41C2, (C22×C4).49(C2×C6), (C3×C4⋊C4).231C22, (C2×C4).103(C22×C6), SmallGroup(192,902)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊2D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >
Subgroups: 282 in 130 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×D8, C22×C12, C6×D4, C6×D4, C8⋊2D4, C3×D4⋊C4, C3×C4.Q8, C3×C4⋊D4, C6×M4(2), C6×D8, C3×C8⋊2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C8⋊C22, C6×D4, C3×C4○D4, C8⋊2D4, C3×C4⋊D4, C3×C8⋊C22, C3×C8⋊2D4
►On 96 points
Generators in S96
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 30 58)(10 31 59)(11 32 60)(12 25 61)(13 26 62)(14 27 63)(15 28 64)(16 29 57)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 54 89)(42 55 90)(43 56 91)(44 49 92)(45 50 93)(46 51 94)(47 52 95)(48 53 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 25 75 43)(2 28 76 46)(3 31 77 41)(4 26 78 44)(5 29 79 47)(6 32 80 42)(7 27 73 45)(8 30 74 48)(9 34 96 24)(10 37 89 19)(11 40 90 22)(12 35 91 17)(13 38 92 20)(14 33 93 23)(15 36 94 18)(16 39 95 21)(49 68 62 86)(50 71 63 81)(51 66 64 84)(52 69 57 87)(53 72 58 82)(54 67 59 85)(55 70 60 88)(56 65 61 83)
(2 8)(3 7)(4 6)(9 94)(10 93)(11 92)(12 91)(13 90)(14 89)(15 96)(16 95)(18 24)(19 23)(20 22)(25 43)(26 42)(27 41)(28 48)(29 47)(30 46)(31 45)(32 44)(33 37)(34 36)(38 40)(49 60)(50 59)(51 58)(52 57)(53 64)(54 63)(55 62)(56 61)(66 72)(67 71)(68 70)(73 77)(74 76)(78 80)(81 85)(82 84)(86 88)
G:=sub<Sym(96)| (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,30,58)(10,31,59)(11,32,60)(12,25,61)(13,26,62)(14,27,63)(15,28,64)(16,29,57)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,54,89)(42,55,90)(43,56,91)(44,49,92)(45,50,93)(46,51,94)(47,52,95)(48,53,96), (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), (1,25,75,43)(2,28,76,46)(3,31,77,41)(4,26,78,44)(5,29,79,47)(6,32,80,42)(7,27,73,45)(8,30,74,48)(9,34,96,24)(10,37,89,19)(11,40,90,22)(12,35,91,17)(13,38,92,20)(14,33,93,23)(15,36,94,18)(16,39,95,21)(49,68,62,86)(50,71,63,81)(51,66,64,84)(52,69,57,87)(53,72,58,82)(54,67,59,85)(55,70,60,88)(56,65,61,83), (2,8)(3,7)(4,6)(9,94)(10,93)(11,92)(12,91)(13,90)(14,89)(15,96)(16,95)(18,24)(19,23)(20,22)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(33,37)(34,36)(38,40)(49,60)(50,59)(51,58)(52,57)(53,64)(54,63)(55,62)(56,61)(66,72)(67,71)(68,70)(73,77)(74,76)(78,80)(81,85)(82,84)(86,88)>;
G:=Group( (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,30,58)(10,31,59)(11,32,60)(12,25,61)(13,26,62)(14,27,63)(15,28,64)(16,29,57)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,54,89)(42,55,90)(43,56,91)(44,49,92)(45,50,93)(46,51,94)(47,52,95)(48,53,96), (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), (1,25,75,43)(2,28,76,46)(3,31,77,41)(4,26,78,44)(5,29,79,47)(6,32,80,42)(7,27,73,45)(8,30,74,48)(9,34,96,24)(10,37,89,19)(11,40,90,22)(12,35,91,17)(13,38,92,20)(14,33,93,23)(15,36,94,18)(16,39,95,21)(49,68,62,86)(50,71,63,81)(51,66,64,84)(52,69,57,87)(53,72,58,82)(54,67,59,85)(55,70,60,88)(56,65,61,83), (2,8)(3,7)(4,6)(9,94)(10,93)(11,92)(12,91)(13,90)(14,89)(15,96)(16,95)(18,24)(19,23)(20,22)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(33,37)(34,36)(38,40)(49,60)(50,59)(51,58)(52,57)(53,64)(54,63)(55,62)(56,61)(66,72)(67,71)(68,70)(73,77)(74,76)(78,80)(81,85)(82,84)(86,88) );
G=PermutationGroup([[(1,65,17),(2,66,18),(3,67,19),(4,68,20),(5,69,21),(6,70,22),(7,71,23),(8,72,24),(9,30,58),(10,31,59),(11,32,60),(12,25,61),(13,26,62),(14,27,63),(15,28,64),(16,29,57),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,54,89),(42,55,90),(43,56,91),(44,49,92),(45,50,93),(46,51,94),(47,52,95),(48,53,96)], [(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)], [(1,25,75,43),(2,28,76,46),(3,31,77,41),(4,26,78,44),(5,29,79,47),(6,32,80,42),(7,27,73,45),(8,30,74,48),(9,34,96,24),(10,37,89,19),(11,40,90,22),(12,35,91,17),(13,38,92,20),(14,33,93,23),(15,36,94,18),(16,39,95,21),(49,68,62,86),(50,71,63,81),(51,66,64,84),(52,69,57,87),(53,72,58,82),(54,67,59,85),(55,70,60,88),(56,65,61,83)], [(2,8),(3,7),(4,6),(9,94),(10,93),(11,92),(12,91),(13,90),(14,89),(15,96),(16,95),(18,24),(19,23),(20,22),(25,43),(26,42),(27,41),(28,48),(29,47),(30,46),(31,45),(32,44),(33,37),(34,36),(38,40),(49,60),(50,59),(51,58),(52,57),(53,64),(54,63),(55,62),(56,61),(66,72),(67,71),(68,70),(73,77),(74,76),(78,80),(81,85),(82,84),(86,88)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×D4 | C3×C4○D4 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C8⋊2D4 | C3×D4⋊C4 | C3×C4.Q8 | C3×C4⋊D4 | C6×M4(2) | C6×D8 | C8⋊2D4 | D4⋊C4 | C4.Q8 | C4⋊D4 | C2×M4(2) | C2×D8 | C24 | C2×C12 | C22×C6 | C12 | C8 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C3×C8⋊2D4 ►in GL8(𝔽73)
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 71 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 50 | 0 | 15 |
| 0 | 0 | 0 | 0 | 59 | 23 | 15 | 58 |
| 0 | 0 | 0 | 0 | 69 | 4 | 0 | 64 |
| 0 | 0 | 0 | 0 | 26 | 51 | 23 | 27 |
| 72 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 56 | 17 | 72 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 2 | 17 | 56 |
| 1 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 1 | 72 |
G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[46,71,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,59,69,26,0,0,0,0,50,23,4,51,0,0,0,0,0,15,0,23,0,0,0,0,15,58,64,27],[72,0,0,0,0,0,0,0,27,1,0,0,0,0,0,0,0,0,72,71,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,56,1,72,0,0,0,0,0,17,0,2,0,0,0,0,1,72,0,17,0,0,0,0,0,2,0,56],[1,0,0,0,0,0,0,0,46,72,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72] >;
C3×C8⋊2D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_2D_4
% in TeX
G:=Group("C3xC8:2D4"); // GroupNames label
G:=SmallGroup(192,902);
// by ID
G=gap.SmallGroup(192,902);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,1059,4204,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations