Copied to
clipboard

G = C24:9D4order 192 = 26·3

9th semidirect product of C24 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24:9D4, C3:C8:10D4, C8:6(C3:D4), C3:5(C8:3D4), C24:C4:6C2, C4.26(S3xD4), (C2xC8).91D6, (C2xD24):25C2, C12:3D4:6C2, (C6xSD16):4C2, (C2xSD16):2S3, (C2xD4).77D6, (C2xQ8).82D6, C12.179(C2xD4), C12.23D4:5C2, C6.32(C4:1D4), C2.31(Q8:3D6), C6.81(C8:C22), (C2xDic3).74D4, C22.273(S3xD4), (C6xQ8).82C22, C2.23(C12:3D4), (C2xC12).453C23, (C2xC24).116C22, (C6xD4).102C22, (C2xD12).123C22, (C4xDic3).52C22, (C2xD4:S3):21C2, C4.10(C2xC3:D4), (C2xC6).365(C2xD4), (C2xQ8:2S3):19C2, (C2xC3:C8).161C22, (C2xC4).542(C22xS3), SmallGroup(192,735)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C24:9D4
C1C3C6C2xC6C2xC12C2xD12C2xD24 — C24:9D4
C3C6C2xC12 — C24:9D4
C1C22C2xC4C2xSD16

Generators and relations for C24:9D4
 G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 536 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C2xC8, C2xC8, D8, SD16, C2xD4, C2xD4, C2xQ8, C3:C8, C24, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C8:C4, C4.4D4, C4:1D4, C2xD8, C2xSD16, C2xSD16, D24, C2xC3:C8, C4xDic3, D6:C4, D4:S3, Q8:2S3, C2xC24, C3xSD16, C2xD12, C2xC3:D4, C6xD4, C6xQ8, C8:3D4, C24:C4, C2xD24, C2xD4:S3, C12:3D4, C2xQ8:2S3, C12.23D4, C6xSD16, C24:9D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C4:1D4, C8:C22, S3xD4, C2xC3:D4, C8:3D4, Q8:3D6, C12:3D4, C24:9D4

Character table of C24:9D4

 class 12A2B2C2D2E2F34A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
 size 111182424222812122228844121244884444
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1111-111111-1-1111111-1-11111    linear of order 2
ρ31111-111111-1-1-1111-1-111-1-111-1-11111    linear of order 2
ρ411111-1-11111-1-11111111-1-111111111    linear of order 2
ρ511111-11111-11111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ61111-11-1111111111-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ71111-1-111111-1-1111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ8111111-1111-1-1-111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ92-2-220002-22000-2-22002-2002-2002-2-22    orthogonal lifted from D4
ρ1022220002-2-20-22222000000-2-2000000    orthogonal lifted from D4
ρ112222-200-122-200-1-1-1112200-1-111-1-1-1-1    orthogonal lifted from D6
ρ122222200-122-200-1-1-1-1-1-2-200-1-1111111    orthogonal lifted from D6
ρ132-2-220002-22000-2-2200-22002-200-222-2    orthogonal lifted from D4
ρ142222200-122200-1-1-1-1-12200-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ152-2-2200022-2000-2-2200002-2-22000000    orthogonal lifted from D4
ρ162222-200-122200-1-1-111-2-200-1-1-1-11111    orthogonal lifted from D6
ρ172-2-2200022-2000-2-220000-22-22000000    orthogonal lifted from D4
ρ1822220002-2-202-2222000000-2-2000000    orthogonal lifted from D4
ρ192-2-22000-1-2200011-1-3--32-200-11--3-3-111-1    complex lifted from C3:D4
ρ202-2-22000-1-2200011-1-3--3-2200-11-3--31-1-11    complex lifted from C3:D4
ρ212-2-22000-1-2200011-1--3-3-2200-11--3-31-1-11    complex lifted from C3:D4
ρ222-2-22000-1-2200011-1--3-32-200-11-3--3-111-1    complex lifted from C3:D4
ρ234-44-4000400000-44-400000000000000    orthogonal lifted from C8:C22
ρ244444000-2-4-4000-2-2-200000022000000    orthogonal lifted from S3xD4
ρ254-4-44000-24-400022-20000002-2000000    orthogonal lifted from S3xD4
ρ2644-4-40004000004-4-400000000000000    orthogonal lifted from C8:C22
ρ2744-4-4000-200000-222000000000066-6-6    orthogonal lifted from Q8:3D6
ρ284-44-4000-2000002-220000000000-66-66    orthogonal lifted from Q8:3D6
ρ294-44-4000-2000002-2200000000006-66-6    orthogonal lifted from Q8:3D6
ρ3044-4-4000-200000-2220000000000-6-666    orthogonal lifted from Q8:3D6

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

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

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

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

Matrix representation of C24:9D4 in GL6(F73)

100000
010000
0034393439
0034683468
0039343439
003953468
,
17700000
48560000
00696570
006946666
006606965
0077694
,
100000
60720000
001000
00727200
0000720
000011

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,39,39,0,0,39,68,34,5,0,0,34,34,34,34,0,0,39,68,39,68],[17,48,0,0,0,0,70,56,0,0,0,0,0,0,69,69,66,7,0,0,65,4,0,7,0,0,7,66,69,69,0,0,0,66,65,4],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;

C24:9D4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_9D_4
% in TeX

G:=Group("C24:9D4");
// GroupNames label

G:=SmallGroup(192,735);
// by ID

G=gap.SmallGroup(192,735);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,1684,438,102,6278]);
// Polycyclic

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

Export

Character table of C24:9D4 in TeX

׿
x
:
Z
F
o
wr
Q
<