direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C24, (C4×C8)⋊4C6, C4⋊C8⋊18C6, C12⋊6(C2×C8), (C4×C24)⋊9C2, C4⋊1(C2×C24), (C22×C8)⋊9C6, C4.79(C6×D4), C2.3(D4×C12), C4⋊C4.11C12, C22⋊C8⋊15C6, C22⋊2(C2×C24), (C22×C24)⋊9C2, (C4×D4).14C6, (C6×D4).23C4, C6.111(C4×D4), (C2×D4).11C12, (D4×C12).29C2, C6.48(C8○D4), C12.484(C2×D4), C22⋊C4.7C12, C42.68(C2×C6), C6.33(C22×C8), C2.4(C22×C24), C23.23(C2×C12), C12.353(C4○D4), (C2×C12).990C23, (C4×C12).353C22, (C2×C24).361C22, C22.22(C22×C12), (C22×C12).499C22, (C2×C6)⋊4(C2×C8), (C3×C4⋊C8)⋊37C2, C2.2(C3×C8○D4), (C3×C4⋊C4).23C4, C4.51(C3×C4○D4), (C3×C22⋊C8)⋊32C2, (C2×C8).107(C2×C6), (C2×C4).36(C2×C12), (C2×C12).212(C2×C4), (C3×C22⋊C4).14C4, (C22×C6).85(C2×C4), (C2×C4).158(C22×C6), (C2×C6).240(C22×C4), (C22×C4).102(C2×C6), SmallGroup(192,867)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C24
G = < a,b,c | a24=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 178 in 134 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C24, C24, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C2×C24, C22×C12, C6×D4, C8×D4, C4×C24, C3×C22⋊C8, C3×C4⋊C8, D4×C12, C22×C24, D4×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C23, C12, C2×C6, C2×C8, C22×C4, C2×D4, C4○D4, C24, C2×C12, C3×D4, C22×C6, C4×D4, C22×C8, C8○D4, C2×C24, C22×C12, C6×D4, C3×C4○D4, C8×D4, D4×C12, C22×C24, C3×C8○D4, D4×C24
►On 96 points
(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 61 80 34)(2 62 81 35)(3 63 82 36)(4 64 83 37)(5 65 84 38)(6 66 85 39)(7 67 86 40)(8 68 87 41)(9 69 88 42)(10 70 89 43)(11 71 90 44)(12 72 91 45)(13 49 92 46)(14 50 93 47)(15 51 94 48)(16 52 95 25)(17 53 96 26)(18 54 73 27)(19 55 74 28)(20 56 75 29)(21 57 76 30)(22 58 77 31)(23 59 78 32)(24 60 79 33)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(49 92)(50 93)(51 94)(52 95)(53 96)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)
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), (1,61,80,34)(2,62,81,35)(3,63,82,36)(4,64,83,37)(5,65,84,38)(6,66,85,39)(7,67,86,40)(8,68,87,41)(9,69,88,42)(10,70,89,43)(11,71,90,44)(12,72,91,45)(13,49,92,46)(14,50,93,47)(15,51,94,48)(16,52,95,25)(17,53,96,26)(18,54,73,27)(19,55,74,28)(20,56,75,29)(21,57,76,30)(22,58,77,31)(23,59,78,32)(24,60,79,33), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(49,92)(50,93)(51,94)(52,95)(53,96)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)>;
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), (1,61,80,34)(2,62,81,35)(3,63,82,36)(4,64,83,37)(5,65,84,38)(6,66,85,39)(7,67,86,40)(8,68,87,41)(9,69,88,42)(10,70,89,43)(11,71,90,44)(12,72,91,45)(13,49,92,46)(14,50,93,47)(15,51,94,48)(16,52,95,25)(17,53,96,26)(18,54,73,27)(19,55,74,28)(20,56,75,29)(21,57,76,30)(22,58,77,31)(23,59,78,32)(24,60,79,33), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(49,92)(50,93)(51,94)(52,95)(53,96)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91) );
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)], [(1,61,80,34),(2,62,81,35),(3,63,82,36),(4,64,83,37),(5,65,84,38),(6,66,85,39),(7,67,86,40),(8,68,87,41),(9,69,88,42),(10,70,89,43),(11,71,90,44),(12,72,91,45),(13,49,92,46),(14,50,93,47),(15,51,94,48),(16,52,95,25),(17,53,96,26),(18,54,73,27),(19,55,74,28),(20,56,75,29),(21,57,76,30),(22,58,77,31),(23,59,78,32),(24,60,79,33)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(49,92),(50,93),(51,94),(52,95),(53,96),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6N | 8A | ··· | 8H | 8I | ··· | 8T | 12A | ··· | 12H | 12I | ··· | 12X | 24A | ··· | 24P | 24Q | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C24 | D4 | C4○D4 | C3×D4 | C8○D4 | C3×C4○D4 | C3×C8○D4 |
| kernel | D4×C24 | C4×C24 | C3×C22⋊C8 | C3×C4⋊C8 | D4×C12 | C22×C24 | C8×D4 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C3×D4 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C24 | C12 | C8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 16 | 8 | 4 | 4 | 32 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of D4×C24 ►in GL3(𝔽73) generated by
| 22 | 0 | 0 |
| 0 | 70 | 0 |
| 0 | 0 | 70 |
| 72 | 0 | 0 |
| 0 | 1 | 19 |
| 0 | 46 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 27 | 1 |
G:=sub<GL(3,GF(73))| [22,0,0,0,70,0,0,0,70],[72,0,0,0,1,46,0,19,72],[1,0,0,0,72,27,0,0,1] >;
D4×C24 in GAP, Magma, Sage, TeX
D_4\times C_{24} % in TeX
G:=Group("D4xC24"); // GroupNames label
G:=SmallGroup(192,867);
// by ID
G=gap.SmallGroup(192,867);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,268,124]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations