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G = D24.1C4order 192 = 26·3

1st non-split extension by D24 of C4 acting via C4/C2=C2

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.C41C2, C4.17(D6⋊C4), (C2×C12).394D4, C32(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

C1C24 — D24.1C4
C1C3C6C12C24C2×C24C4○D24 — D24.1C4
C3C6C12C24 — D24.1C4
C1C4C2×C4C2×C8C2×C16

Generators and relations for D24.1C4
 G = < a,b,c | a24=b2=1, c4=a18, bab=a-1, ac=ca, cbc-1=a15b >

2C2
24C2
12C4
12C22
2C6
8S3
6Q8
6D4
12D4
12C2×C4
12C8
4D6
4Dic3
2C16
3D8
3Q16
6M4(2)
6SD16
6C4○D4
2Dic6
2D12
4C3⋊C8
4C3⋊D4
4C4×S3
3C8.C4
3C4○D8
2C48
2C4○D12
2C24⋊C2
2C4.Dic3
3D8.C4

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 30)(26 29)(27 28)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 72)(65 71)(66 70)(67 69)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(88 96)(89 95)(90 94)(91 93)
(1 52 43 91 19 70 37 85 13 64 31 79 7 58 25 73)(2 53 44 92 20 71 38 86 14 65 32 80 8 59 26 74)(3 54 45 93 21 72 39 87 15 66 33 81 9 60 27 75)(4 55 46 94 22 49 40 88 16 67 34 82 10 61 28 76)(5 56 47 95 23 50 41 89 17 68 35 83 11 62 29 77)(6 57 48 96 24 51 42 90 18 69 36 84 12 63 30 78)

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)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,96)(89,95)(90,94)(91,93), (1,52,43,91,19,70,37,85,13,64,31,79,7,58,25,73)(2,53,44,92,20,71,38,86,14,65,32,80,8,59,26,74)(3,54,45,93,21,72,39,87,15,66,33,81,9,60,27,75)(4,55,46,94,22,49,40,88,16,67,34,82,10,61,28,76)(5,56,47,95,23,50,41,89,17,68,35,83,11,62,29,77)(6,57,48,96,24,51,42,90,18,69,36,84,12,63,30,78)>;

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)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,96)(89,95)(90,94)(91,93), (1,52,43,91,19,70,37,85,13,64,31,79,7,58,25,73)(2,53,44,92,20,71,38,86,14,65,32,80,8,59,26,74)(3,54,45,93,21,72,39,87,15,66,33,81,9,60,27,75)(4,55,46,94,22,49,40,88,16,67,34,82,10,61,28,76)(5,56,47,95,23,50,41,89,17,68,35,83,11,62,29,77)(6,57,48,96,24,51,42,90,18,69,36,84,12,63,30,78) );

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),(25,30),(26,29),(27,28),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(64,72),(65,71),(66,70),(67,69),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(88,96),(89,95),(90,94),(91,93)], [(1,52,43,91,19,70,37,85,13,64,31,79,7,58,25,73),(2,53,44,92,20,71,38,86,14,65,32,80,8,59,26,74),(3,54,45,93,21,72,39,87,15,66,33,81,9,60,27,75),(4,55,46,94,22,49,40,88,16,67,34,82,10,61,28,76),(5,56,47,95,23,50,41,89,17,68,35,83,11,62,29,77),(6,57,48,96,24,51,42,90,18,69,36,84,12,63,30,78)])

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D6A6B6C8A8B8C8D8E8F12A12B12C12D16A···16H24A···24H48A···48P
order1222344446668888881212121216···1624···2448···48
size112242112242222222242422222···22···22···2

54 irreducible representations

dim1111112222222222222
type+++++++++++
imageC1C2C2C2C4C4S3D4D4D6D8SD16C4×S3C3⋊D4D12D24C24⋊C2D8.C4D24.1C4
kernelD24.1C4C24.C4C2×C48C4○D24D24Dic12C2×C16C24C2×C12C2×C8C12C2×C6C8C8C2×C4C4C22C3C1
# reps11112211112222244816

Matrix representation of D24.1C4 in GL2(𝔽97) generated by

162
9518
,
9518
162
,
4820
7768
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

Export

Subgroup lattice of D24.1C4 in TeX

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