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G = C24.43D4order 192 = 26·3

43rd non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.43D4, C3⋊C8.20D4, C4.24(S3×D4), (C6×SD16)⋊7C2, (C2×SD16)⋊9S3, (C2×D4).71D6, (C2×C8).262D6, (C2×Q8).77D6, (C8×Dic3)⋊10C2, C6.62(C4○D8), C12.175(C2×D4), C8.20(C3⋊D4), C34(C8.12D4), C23.12D66C2, C12.23D44C2, C6.30(C41D4), (C6×D4).94C22, C22.265(S3×D4), (C6×Q8).75C22, C2.21(C123D4), (C2×C12).445C23, (C2×C24).163C22, (C2×Dic3).113D4, C2.28(Q8.7D6), (C2×D12).119C22, (C2×Dic6).126C22, (C4×Dic3).241C22, C4.8(C2×C3⋊D4), (C2×D4⋊S3).9C2, (C2×C24⋊C2)⋊29C2, (C2×C3⋊Q16)⋊18C2, (C2×C6).357(C2×D4), (C2×C3⋊C8).273C22, (C2×C4).534(C22×S3), SmallGroup(192,727)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.43D4
C1C3C6C12C2×C12C4×Dic3C23.12D6 — C24.43D4
C3C6C2×C12 — C24.43D4
C1C22C2×C4C2×SD16

Generators and relations for C24.43D4
 G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a11, cbc-1=a12b-1 >

Subgroups: 408 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3, C6, C6 [×2], C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×3], C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], C24 [×2], Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16, C2×SD16, C2×Q16, C24⋊C2 [×2], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×2], D4⋊S3 [×2], C3⋊Q16 [×2], C6.D4 [×2], C2×C24, C3×SD16 [×2], C2×Dic6, C2×D12, C6×D4, C6×Q8, C8.12D4, C8×Dic3, C2×C24⋊C2, C2×D4⋊S3, C23.12D6, C2×C3⋊Q16, C12.23D4, C6×SD16, C24.43D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C4○D8 [×2], S3×D4 [×2], C2×C3⋊D4, C8.12D4, Q8.7D6 [×2], C123D4, C24.43D4

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

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222234444444466666888888881212121224242424
size11118242226666824222882222666644884444

36 irreducible representations

dim11111111222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6D6C3⋊D4C4○D8S3×D4S3×D4Q8.7D6
kernelC24.43D4C8×Dic3C2×C24⋊C2C2×D4⋊S3C23.12D6C2×C3⋊Q16C12.23D4C6×SD16C2×SD16C3⋊C8C24C2×Dic3C2×C8C2×D4C2×Q8C8C6C4C22C2
# reps11111111122211148114

Matrix representation of C24.43D4 in GL6(𝔽73)

6760000
67670000
000100
0072100
00005371
00001820
,
0460000
2700000
000100
001000
0000202
00005553
,
6670000
67670000
0007200
0072000
00005371
00001720

G:=sub<GL(6,GF(73))| [67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,53,18,0,0,0,0,71,20],[0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,55,0,0,0,0,2,53],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,53,17,0,0,0,0,71,20] >;

C24.43D4 in GAP, Magma, Sage, TeX

C_{24}._{43}D_4
% in TeX

G:=Group("C24.43D4");
// GroupNames label

G:=SmallGroup(192,727);
// by ID

G=gap.SmallGroup(192,727);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,184,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^17,c*a*c^-1=a^11,c*b*c^-1=a^12*b^-1>;
// generators/relations

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