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G = C24.99D4order 192 = 26·3

22nd non-split extension by C24 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.99D4, M4(2).3Dic3, D4.(C3⋊C8), C12.9(C2×C8), C33(D4.C8), C8○D4.4S3, (C3×D4).1C8, Q8.2(C3⋊C8), (C3×Q8).1C8, (C2×C8).268D6, C8.34(C3⋊D4), C4○D4.4Dic3, C12.C812C2, (C2×C6).6M4(2), C6.19(C22⋊C8), (C3×M4(2)).5C4, C12.96(C22⋊C4), (C2×C24).266C22, C2.8(C12.55D4), C4.30(C6.D4), C22.1(C4.Dic3), C4.3(C2×C3⋊C8), (C2×C3⋊C16)⋊14C2, (C3×C4○D4).1C4, (C3×C8○D4).3C2, (C2×C12).67(C2×C4), (C2×C4).41(C2×Dic3), SmallGroup(192,120)

Series: Derived Chief Lower central Upper central

C1C12 — C24.99D4
C1C3C6C12C24C2×C24C12.C8 — C24.99D4
C3C6C12 — C24.99D4
C1C8C2×C8C8○D4

Generators and relations for C24.99D4
 G = < a,b,c | a24=1, b4=a6, c2=a9, bab-1=cac-1=a17, cbc-1=a3b3 >

2C2
4C2
2C4
2C22
2C6
4C6
2D4
2C8
2C2×C4
2C2×C6
2C12
2M4(2)
2C2×C8
6C16
6C16
2C24
2C3×D4
2C2×C12
3C2×C16
3M5(2)
2C3⋊C16
2C3⋊C16
2C2×C24
2C3×M4(2)
3D4.C8

Smallest permutation representation of C24.99D4
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 75 66 39 7 81 72 45 13 87 54 27 19 93 60 33)(2 92 67 32 8 74 49 38 14 80 55 44 20 86 61 26)(3 85 68 25 9 91 50 31 15 73 56 37 21 79 62 43)(4 78 69 42 10 84 51 48 16 90 57 30 22 96 63 36)(5 95 70 35 11 77 52 41 17 83 58 47 23 89 64 29)(6 88 71 28 12 94 53 34 18 76 59 40 24 82 65 46)
(1 42 10 27 19 36 4 45 13 30 22 39 7 48 16 33)(2 35 11 44 20 29 5 38 14 47 23 32 8 41 17 26)(3 28 12 37 21 46 6 31 15 40 24 25 9 34 18 43)(49 95 58 80 67 89 52 74 61 83 70 92 55 77 64 86)(50 88 59 73 68 82 53 91 62 76 71 85 56 94 65 79)(51 81 60 90 69 75 54 84 63 93 72 78 57 87 66 96)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D6A6B6C6D8A8B8C8D8E8F8G8H12A12B12C12D12E16A···16H16I16J16K16L24A24B24C24D24E···24J
order122234444666688888888121212121216···16161616162424242424···24
size112421124244411112244224446···61212121222224···4

48 irreducible representations

dim11111111222222222224
type+++++++--
imageC1C2C2C2C4C4C8C8S3D4D6Dic3Dic3M4(2)C3⋊D4C3⋊C8C3⋊C8C4.Dic3D4.C8C24.99D4
kernelC24.99D4C2×C3⋊C16C12.C8C3×C8○D4C3×M4(2)C3×C4○D4C3×D4C3×Q8C8○D4C24C2×C8M4(2)C4○D4C2×C6C8D4Q8C22C3C1
# reps11112244121112422484

Matrix representation of C24.99D4 in GL4(𝔽97) generated by

33000
03300
00350
00061
,
03900
293900
0001
00960
,
03900
68000
0001
0010
G:=sub<GL(4,GF(97))| [33,0,0,0,0,33,0,0,0,0,35,0,0,0,0,61],[0,29,0,0,39,39,0,0,0,0,0,96,0,0,1,0],[0,68,0,0,39,0,0,0,0,0,0,1,0,0,1,0] >;

C24.99D4 in GAP, Magma, Sage, TeX

C_{24}._{99}D_4
% in TeX

G:=Group("C24.99D4");
// GroupNames label

G:=SmallGroup(192,120);
// by ID

G=gap.SmallGroup(192,120);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,100,1123,570,136,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^3*b^3>;
// generators/relations

Export

Subgroup lattice of C24.99D4 in TeX

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