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G = C24.31D4order 192 = 26·3

31st non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.31D4, C3:C8.9D4, C24:C4:5C2, C4.23(S3xD4), (C2xC8).89D6, (C2xD4).70D6, C8.3(C3:D4), C3:4(C8.2D4), (C2xQ8).76D6, C12.174(C2xD4), (C2xSD16).2S3, (C6xSD16).2C2, Dic3:Q8:4C2, (C2xDic12):25C2, C6.29(C4:1D4), (C2xDic3).70D4, (C6xD4).93C22, C22.264(S3xD4), (C6xQ8).74C22, C2.20(C12:3D4), (C2xC12).444C23, (C2xC24).114C22, C23.12D6.6C2, C2.28(D4.D6), C6.48(C8.C22), (C4xDic3).51C22, (C2xDic6).125C22, C4.7(C2xC3:D4), (C2xC3:Q16):17C2, (C2xC6).356(C2xD4), (C2xD4.S3).9C2, (C2xC3:C8).156C22, (C2xC4).533(C22xS3), SmallGroup(192,726)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C24.31D4
C1C3C6C12C2xC12C4xDic3Dic3:Q8 — C24.31D4
C3C6C2xC12 — C24.31D4
C1C22C2xC4C2xSD16

Generators and relations for C24.31D4
 G = < a,b,c | a24=b4=1, c2=a12, bab-1=a5, cac-1=a-1, cbc-1=b-1 >

Subgroups: 344 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, SD16, Q16, C2xD4, C2xQ8, C2xQ8, C3:C8, C24, Dic6, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C8:C4, C4.4D4, C4:Q8, C2xSD16, C2xSD16, C2xQ16, Dic12, C2xC3:C8, C4xDic3, Dic3:C4, D4.S3, C3:Q16, C6.D4, C2xC24, C3xSD16, C2xDic6, C6xD4, C6xQ8, C8.2D4, C24:C4, C2xDic12, C2xD4.S3, C23.12D6, C2xC3:Q16, Dic3:Q8, C6xSD16, C24.31D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4:1D4, C8.C22, S3xD4, C2xC3:D4, C8.2D4, D4.D6, C12:3D4, C24.31D4

Character table of C24.31D4

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
 size 111182228121224242228844121244884444
ρ1111111111111111111111111111111    trivial
ρ21111-1111-1-1-111111-1-111-1-111-1-11111    linear of order 2
ρ31111-11111111-1111-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ411111111-1-1-11-111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ5111111111-1-1-1-11111111-1-111111111    linear of order 2
ρ61111-1111-111-1-1111-1-1111111-1-11111    linear of order 2
ρ71111-11111-1-1-11111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ811111111-111-1111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ92222-2-12220000-1-1-111-2-200-1-1-1-11111    orthogonal lifted from D6
ρ102222-2-122-20000-1-1-1112200-1-111-1-1-1-1    orthogonal lifted from D6
ρ112-2-2202-2200000-2-2200-22002-20022-2-2    orthogonal lifted from D4
ρ12222202-2-202-200222000000-2-2000000    orthogonal lifted from D4
ρ13222202-2-20-2200222000000-2-2000000    orthogonal lifted from D4
ρ142-2-2202-2200000-2-22002-2002-200-2-222    orthogonal lifted from D4
ρ1522222-12220000-1-1-1-1-12200-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622222-122-20000-1-1-1-1-1-2-200-1-1111111    orthogonal lifted from D6
ρ172-2-22022-200000-2-220000-22-22000000    orthogonal lifted from D4
ρ182-2-22022-200000-2-2200002-2-22000000    orthogonal lifted from D4
ρ192-2-220-1-220000011-1--3-32-200-11--3-311-1-1    complex lifted from C3:D4
ρ202-2-220-1-220000011-1--3-3-2200-11-3--3-1-111    complex lifted from C3:D4
ρ212-2-220-1-220000011-1-3--3-2200-11--3-3-1-111    complex lifted from C3:D4
ρ222-2-220-1-220000011-1-3--32-200-11-3--311-1-1    complex lifted from C3:D4
ρ234-4-440-24-40000022-20000002-2000000    orthogonal lifted from S3xD4
ρ2444440-2-4-400000-2-2-200000022000000    orthogonal lifted from S3xD4
ρ254-44-4040000000-44-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-40400000004-4-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-44-40-200000002-2200000000006-6-66    symplectic lifted from D4.D6, Schur index 2
ρ2844-4-40-20000000-2220000000000-66-66    symplectic lifted from D4.D6, Schur index 2
ρ2944-4-40-20000000-22200000000006-66-6    symplectic lifted from D4.D6, Schur index 2
ρ304-44-40-200000002-220000000000-666-6    symplectic lifted from D4.D6, Schur index 2

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

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

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

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

Matrix representation of C24.31D4 in GL8(F73)

68548500000
686348480000
13710680000
371550000
000000685
0000006863
00003934568
0000395510
,
67843700000
14670330000
562565670000
25425980000
000048112322
000036257250
000025622562
000037483748
,
5550000000
6818000000
0050180000
0068230000
0000001613
0000007057
000084300
0000356500

G:=sub<GL(8,GF(73))| [68,68,1,37,0,0,0,0,5,63,37,1,0,0,0,0,48,48,10,5,0,0,0,0,50,48,68,5,0,0,0,0,0,0,0,0,0,0,39,39,0,0,0,0,0,0,34,5,0,0,0,0,68,68,5,5,0,0,0,0,5,63,68,10],[67,14,56,25,0,0,0,0,8,6,25,42,0,0,0,0,43,70,65,59,0,0,0,0,70,33,67,8,0,0,0,0,0,0,0,0,48,36,25,37,0,0,0,0,11,25,62,48,0,0,0,0,23,72,25,37,0,0,0,0,22,50,62,48],[55,68,0,0,0,0,0,0,50,18,0,0,0,0,0,0,0,0,50,68,0,0,0,0,0,0,18,23,0,0,0,0,0,0,0,0,0,0,8,35,0,0,0,0,0,0,43,65,0,0,0,0,16,70,0,0,0,0,0,0,13,57,0,0] >;

C24.31D4 in GAP, Magma, Sage, TeX

C_{24}._{31}D_4
% in TeX

G:=Group("C24.31D4");
// GroupNames label

G:=SmallGroup(192,726);
// by ID

G=gap.SmallGroup(192,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,1094,135,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Character table of C24.31D4 in TeX

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