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G = C24.82D4order 192 = 26·3

5th non-split extension by C24 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.82D4, C222Dic12, C23.31D12, (C2×C6)⋊5Q16, C241C415C2, (C2×C4).69D12, (C2×C8).309D6, C6.11(C2×Q16), (C2×Dic12)⋊9C2, C6.19(C4○D8), (C2×C12).357D4, C12.414(C2×D4), C8.39(C3⋊D4), C34(C8.18D4), (C22×C8).11S3, C2.Dic123C2, C2.19(C4○D24), C6.72(C4⋊D4), (C22×C24).15C2, C2.11(C2×Dic12), (C22×C4).448D6, (C22×C6).142D4, C4.113(C4○D12), C12.229(C4○D4), C2.20(C127D4), (C2×C12).770C23, (C2×C24).381C22, C12.48D4.5C2, C22.133(C2×D12), C4⋊Dic3.25C22, (C2×Dic6).18C22, (C22×C12).520C22, (C2×C6).160(C2×D4), C4.107(C2×C3⋊D4), (C2×C4).718(C22×S3), SmallGroup(192,675)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.82D4
C1C3C6C2×C6C2×C12C2×Dic6C2×Dic12 — C24.82D4
C3C6C2×C12 — C24.82D4
C1C22C22×C4C22×C8

Generators and relations for C24.82D4
 G = < a,b,c | a24=b4=1, c2=a12, bab-1=cac-1=a-1, cbc-1=a12b-1 >

Subgroups: 296 in 114 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C24, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, Dic12, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C2×Dic6, C22×C12, C8.18D4, C2.Dic12, C241C4, C2×Dic12, C12.48D4, C22×C24, C24.82D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×Q16, C4○D8, Dic12, C2×D12, C4○D12, C2×C3⋊D4, C8.18D4, C4○D24, C2×Dic12, C127D4, C24.82D4

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A···6G8A···8H12A···12H24A···24P
order1222223444444446···68···812···1224···24
size11112222222242424242···22···22···22···2

54 irreducible representations

dim111111222222222222222
type++++++++++++-++-
imageC1C2C2C2C2C2S3D4D4D4D6D6C4○D4Q16C3⋊D4D12D12C4○D8C4○D12Dic12C4○D24
kernelC24.82D4C2.Dic12C241C4C2×Dic12C12.48D4C22×C24C22×C8C24C2×C12C22×C6C2×C8C22×C4C12C2×C6C8C2×C4C23C6C4C22C2
# reps121121121121244224488

Matrix representation of C24.82D4 in GL4(𝔽73) generated by

21000
0700
00700
00024
,
0100
1000
0001
00720
,
0100
72000
0001
0010
G:=sub<GL(4,GF(73))| [21,0,0,0,0,7,0,0,0,0,70,0,0,0,0,24],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,1,0],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C24.82D4 in GAP, Magma, Sage, TeX

C_{24}._{82}D_4
% in TeX

G:=Group("C24.82D4");
// GroupNames label

G:=SmallGroup(192,675);
// by ID

G=gap.SmallGroup(192,675);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,254,1684,102,6278]);
// Polycyclic

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

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