direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊D4, C24⋊16D4, C8⋊1(C3×D4), C2.D8⋊12C6, C4.60(C6×D4), C22⋊Q8⋊4C6, (C2×SD16)⋊1C6, C4⋊D4.4C6, D4⋊C4⋊17C6, Q8⋊C4⋊17C6, (C6×SD16)⋊12C2, C12.467(C2×D4), (C2×C12).327D4, (C2×M4(2))⋊1C6, (C6×M4(2))⋊6C2, C22.92(C6×D4), C23.15(C3×D4), (C22×C6).33D4, C12.265(C4○D4), C6.151(C4⋊D4), C6.137(C8⋊C22), (C2×C12).927C23, (C2×C24).332C22, (C6×D4).191C22, (C6×Q8).165C22, C6.137(C8.C22), (C22×C12).425C22, C4⋊C4.8(C2×C6), (C2×C8).21(C2×C6), (C3×C2.D8)⋊27C2, C4.10(C3×C4○D4), (C2×C4).32(C3×D4), (C2×D4).14(C2×C6), (C2×C6).648(C2×D4), C2.20(C3×C4⋊D4), C2.12(C3×C8⋊C22), (C3×C22⋊Q8)⋊31C2, (C2×Q8).10(C2×C6), (C3×D4⋊C4)⋊40C2, (C3×Q8⋊C4)⋊40C2, (C3×C4⋊D4).14C2, (C22×C4).48(C2×C6), C2.12(C3×C8.C22), (C3×C4⋊C4).230C22, (C2×C4).102(C22×C6), SmallGroup(192,901)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b3, dcd=c-1 >
Subgroups: 234 in 120 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×SD16, C22×C12, C6×D4, C6×D4, C6×Q8, C8⋊D4, C3×D4⋊C4, C3×Q8⋊C4, C3×C2.D8, C3×C4⋊D4, C3×C22⋊Q8, C6×M4(2), C6×SD16, C3×C8⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C8⋊C22, C8.C22, C6×D4, C3×C4○D4, C8⋊D4, C3×C4⋊D4, C3×C8⋊C22, C3×C8.C22, C3×C8⋊D4
►On 96 points
(1 87 37)(2 88 38)(3 81 39)(4 82 40)(5 83 33)(6 84 34)(7 85 35)(8 86 36)(9 19 59)(10 20 60)(11 21 61)(12 22 62)(13 23 63)(14 24 64)(15 17 57)(16 18 58)(25 67 80)(26 68 73)(27 69 74)(28 70 75)(29 71 76)(30 72 77)(31 65 78)(32 66 79)(41 51 91)(42 52 92)(43 53 93)(44 54 94)(45 55 95)(46 56 96)(47 49 89)(48 50 90)
(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 45 13 75)(2 44 14 74)(3 43 15 73)(4 42 16 80)(5 41 9 79)(6 48 10 78)(7 47 11 77)(8 46 12 76)(17 26 81 53)(18 25 82 52)(19 32 83 51)(20 31 84 50)(21 30 85 49)(22 29 86 56)(23 28 87 55)(24 27 88 54)(33 91 59 66)(34 90 60 65)(35 89 61 72)(36 96 62 71)(37 95 63 70)(38 94 64 69)(39 93 57 68)(40 92 58 67)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 54)(26 49)(27 52)(28 55)(29 50)(30 53)(31 56)(32 51)(34 36)(35 39)(38 40)(41 79)(42 74)(43 77)(44 80)(45 75)(46 78)(47 73)(48 76)(57 61)(58 64)(60 62)(65 96)(66 91)(67 94)(68 89)(69 92)(70 95)(71 90)(72 93)(81 85)(82 88)(84 86)
G:=sub<Sym(96)| (1,87,37)(2,88,38)(3,81,39)(4,82,40)(5,83,33)(6,84,34)(7,85,35)(8,86,36)(9,19,59)(10,20,60)(11,21,61)(12,22,62)(13,23,63)(14,24,64)(15,17,57)(16,18,58)(25,67,80)(26,68,73)(27,69,74)(28,70,75)(29,71,76)(30,72,77)(31,65,78)(32,66,79)(41,51,91)(42,52,92)(43,53,93)(44,54,94)(45,55,95)(46,56,96)(47,49,89)(48,50,90), (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), (1,45,13,75)(2,44,14,74)(3,43,15,73)(4,42,16,80)(5,41,9,79)(6,48,10,78)(7,47,11,77)(8,46,12,76)(17,26,81,53)(18,25,82,52)(19,32,83,51)(20,31,84,50)(21,30,85,49)(22,29,86,56)(23,28,87,55)(24,27,88,54)(33,91,59,66)(34,90,60,65)(35,89,61,72)(36,96,62,71)(37,95,63,70)(38,94,64,69)(39,93,57,68)(40,92,58,67), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,54)(26,49)(27,52)(28,55)(29,50)(30,53)(31,56)(32,51)(34,36)(35,39)(38,40)(41,79)(42,74)(43,77)(44,80)(45,75)(46,78)(47,73)(48,76)(57,61)(58,64)(60,62)(65,96)(66,91)(67,94)(68,89)(69,92)(70,95)(71,90)(72,93)(81,85)(82,88)(84,86)>;
G:=Group( (1,87,37)(2,88,38)(3,81,39)(4,82,40)(5,83,33)(6,84,34)(7,85,35)(8,86,36)(9,19,59)(10,20,60)(11,21,61)(12,22,62)(13,23,63)(14,24,64)(15,17,57)(16,18,58)(25,67,80)(26,68,73)(27,69,74)(28,70,75)(29,71,76)(30,72,77)(31,65,78)(32,66,79)(41,51,91)(42,52,92)(43,53,93)(44,54,94)(45,55,95)(46,56,96)(47,49,89)(48,50,90), (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), (1,45,13,75)(2,44,14,74)(3,43,15,73)(4,42,16,80)(5,41,9,79)(6,48,10,78)(7,47,11,77)(8,46,12,76)(17,26,81,53)(18,25,82,52)(19,32,83,51)(20,31,84,50)(21,30,85,49)(22,29,86,56)(23,28,87,55)(24,27,88,54)(33,91,59,66)(34,90,60,65)(35,89,61,72)(36,96,62,71)(37,95,63,70)(38,94,64,69)(39,93,57,68)(40,92,58,67), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,21)(18,24)(20,22)(25,54)(26,49)(27,52)(28,55)(29,50)(30,53)(31,56)(32,51)(34,36)(35,39)(38,40)(41,79)(42,74)(43,77)(44,80)(45,75)(46,78)(47,73)(48,76)(57,61)(58,64)(60,62)(65,96)(66,91)(67,94)(68,89)(69,92)(70,95)(71,90)(72,93)(81,85)(82,88)(84,86) );
G=PermutationGroup([[(1,87,37),(2,88,38),(3,81,39),(4,82,40),(5,83,33),(6,84,34),(7,85,35),(8,86,36),(9,19,59),(10,20,60),(11,21,61),(12,22,62),(13,23,63),(14,24,64),(15,17,57),(16,18,58),(25,67,80),(26,68,73),(27,69,74),(28,70,75),(29,71,76),(30,72,77),(31,65,78),(32,66,79),(41,51,91),(42,52,92),(43,53,93),(44,54,94),(45,55,95),(46,56,96),(47,49,89),(48,50,90)], [(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)], [(1,45,13,75),(2,44,14,74),(3,43,15,73),(4,42,16,80),(5,41,9,79),(6,48,10,78),(7,47,11,77),(8,46,12,76),(17,26,81,53),(18,25,82,52),(19,32,83,51),(20,31,84,50),(21,30,85,49),(22,29,86,56),(23,28,87,55),(24,27,88,54),(33,91,59,66),(34,90,60,65),(35,89,61,72),(36,96,62,71),(37,95,63,70),(38,94,64,69),(39,93,57,68),(40,92,58,67)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,21),(18,24),(20,22),(25,54),(26,49),(27,52),(28,55),(29,50),(30,53),(31,56),(32,51),(34,36),(35,39),(38,40),(41,79),(42,74),(43,77),(44,80),(45,75),(46,78),(47,73),(48,76),(57,61),(58,64),(60,62),(65,96),(66,91),(67,94),(68,89),(69,92),(70,95),(71,90),(72,93),(81,85),(82,88),(84,86)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
48 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×D4 | C3×C4○D4 | C8⋊C22 | C8.C22 | C3×C8⋊C22 | C3×C8.C22 |
| kernel | C3×C8⋊D4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C2.D8 | C3×C4⋊D4 | C3×C22⋊Q8 | C6×M4(2) | C6×SD16 | C8⋊D4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4⋊D4 | C22⋊Q8 | C2×M4(2) | C2×SD16 | C24 | C2×C12 | C22×C6 | C12 | C8 | C2×C4 | C23 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C8⋊D4 ►in GL8(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 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 |
| 52 | 54 | 0 | 0 | 0 | 0 | 0 | 0 |
| 54 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 0 | 27 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 27 | 46 | 0 | 46 |
| 17 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 71 | 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 1 |
| 1 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 1 |
G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[52,54,0,0,0,0,0,0,54,21,0,0,0,0,0,0,0,0,52,1,0,0,0,0,0,0,69,21,0,0,0,0,0,0,0,0,0,0,27,27,0,0,0,0,0,27,0,46,0,0,0,0,46,0,0,0,0,0,0,0,27,27,46,46],[17,71,0,0,0,0,0,0,72,56,0,0,0,0,0,0,0,0,1,26,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,1,0,72,0,0,0,0,0,71,0,1],[1,0,0,0,0,0,0,0,17,72,0,0,0,0,0,0,0,0,1,26,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1] >;
C3×C8⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes D_4
% in TeX
G:=Group("C3xC8:D4"); // GroupNames label
G:=SmallGroup(192,901);
// by ID
G=gap.SmallGroup(192,901);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,1059,4204,172]);
// 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^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations