metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24.1C4, C24.85D4, C4.19D24, C12.37D8, Dic12.1C4, (C2×C48)⋊4C2, (C2×C16)⋊4S3, C8.20(C4×S3), C24.50(C2×C4), C4○D24.1C2, (C2×C4).75D12, (C2×C8).312D6, C24.C4⋊1C2, C4.17(D6⋊C4), (C2×C12).394D4, C3⋊2(D8.C4), C8.42(C3⋊D4), (C2×C6).18SD16, C2.8(C2.D24), C6.16(D4⋊C4), C12.41(C22⋊C4), (C2×C24).384C22, C22.1(C24⋊C2), SmallGroup(192,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24.1C4
G = < a,b,c | a24=b2=1, c4=a18, bab=a-1, ac=ca, cbc-1=a15b >
Smallest permutation representation of D24.1C4
►On 96 points
Generators in S96
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(40 48)(41 47)(42 46)(43 45)(49 59)(50 58)(51 57)(52 56)(53 55)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)(73 80)(74 79)(75 78)(76 77)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)
(1 62 92 31 19 56 86 25 13 50 80 43 7 68 74 37)(2 63 93 32 20 57 87 26 14 51 81 44 8 69 75 38)(3 64 94 33 21 58 88 27 15 52 82 45 9 70 76 39)(4 65 95 34 22 59 89 28 16 53 83 46 10 71 77 40)(5 66 96 35 23 60 90 29 17 54 84 47 11 72 78 41)(6 67 73 36 24 61 91 30 18 55 85 48 12 49 79 42)
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(40,48)(41,47)(42,46)(43,45)(49,59)(50,58)(51,57)(52,56)(53,55)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(73,80)(74,79)(75,78)(76,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89), (1,62,92,31,19,56,86,25,13,50,80,43,7,68,74,37)(2,63,93,32,20,57,87,26,14,51,81,44,8,69,75,38)(3,64,94,33,21,58,88,27,15,52,82,45,9,70,76,39)(4,65,95,34,22,59,89,28,16,53,83,46,10,71,77,40)(5,66,96,35,23,60,90,29,17,54,84,47,11,72,78,41)(6,67,73,36,24,61,91,30,18,55,85,48,12,49,79,42)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(40,48)(41,47)(42,46)(43,45)(49,59)(50,58)(51,57)(52,56)(53,55)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(73,80)(74,79)(75,78)(76,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89), (1,62,92,31,19,56,86,25,13,50,80,43,7,68,74,37)(2,63,93,32,20,57,87,26,14,51,81,44,8,69,75,38)(3,64,94,33,21,58,88,27,15,52,82,45,9,70,76,39)(4,65,95,34,22,59,89,28,16,53,83,46,10,71,77,40)(5,66,96,35,23,60,90,29,17,54,84,47,11,72,78,41)(6,67,73,36,24,61,91,30,18,55,85,48,12,49,79,42) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(40,48),(41,47),(42,46),(43,45),(49,59),(50,58),(51,57),(52,56),(53,55),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67),(73,80),(74,79),(75,78),(76,77),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89)], [(1,62,92,31,19,56,86,25,13,50,80,43,7,68,74,37),(2,63,93,32,20,57,87,26,14,51,81,44,8,69,75,38),(3,64,94,33,21,58,88,27,15,52,82,45,9,70,76,39),(4,65,95,34,22,59,89,28,16,53,83,46,10,71,77,40),(5,66,96,35,23,60,90,29,17,54,84,47,11,72,78,41),(6,67,73,36,24,61,91,30,18,55,85,48,12,49,79,42)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 24 | 2 | 1 | 1 | 2 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | C3⋊D4 | D12 | D24 | C24⋊C2 | D8.C4 | D24.1C4 |
| kernel | D24.1C4 | C24.C4 | C2×C48 | C4○D24 | D24 | Dic12 | C2×C16 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of D24.1C4 ►in GL2(𝔽97) generated by
| 16 | 2 |
| 95 | 18 |
| 95 | 18 |
| 16 | 2 |
| 48 | 20 |
| 77 | 68 |
G:=sub<GL(2,GF(97))| [16,95,2,18],[95,16,18,2],[48,77,20,68] >;
D24.1C4 in GAP, Magma, Sage, TeX
D_{24}._1C_4 % in TeX
G:=Group("D24.1C4"); // GroupNames label
G:=SmallGroup(192,69);
// by ID
G=gap.SmallGroup(192,69);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,268,1123,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=1,c^4=a^18,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations
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