Copied to
clipboard

G = C323Q32order 288 = 25·32

2nd semidirect product of C32 and Q32 acting via Q32/C8=C22

metabelian, supersoluble, monomial

Aliases: C323Q32, C32Dic24, C6.15D24, C24.50D6, C12.12D12, Dic12.1S3, C8.7S32, C3⋊C16.S3, (C3×C6).12D8, C6.4(D4⋊S3), (C3×C12).27D4, C31(C3⋊Q32), C4.4(C3⋊D12), C2.7(C3⋊D24), C325Q16.2C2, C12.69(C3⋊D4), (C3×C24).12C22, (C3×Dic12).3C2, (C3×C3⋊C16).1C2, SmallGroup(288,199)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C323Q32
C1C3C32C3×C6C3×C12C3×C24C3×Dic12 — C323Q32
C32C3×C6C3×C12C3×C24 — C323Q32
C1C2C4C8

Generators and relations for C323Q32
 G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 258 in 57 conjugacy classes, 22 normal (all characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C6 [×2], C6, C8, Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×2], C16, Q16 [×2], C3×C6, C24 [×2], C24, Dic6 [×5], C3×Q8, Q32, C3×Dic3, C3⋊Dic3, C3×C12, C3⋊C16, C48, Dic12, Dic12 [×3], C3×Q16, C3×C24, C3×Dic6, C324Q8, Dic24, C3⋊Q32, C3×C3⋊C16, C3×Dic12, C325Q16, C323Q32
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, D12, C3⋊D4, Q32, S32, D24, D4⋊S3, C3⋊D12, Dic24, C3⋊Q32, C3⋊D24, C323Q32

Smallest permutation representation of C323Q32
On 96 points
Generators in S96
(1 41 54)(2 55 42)(3 43 56)(4 57 44)(5 45 58)(6 59 46)(7 47 60)(8 61 48)(9 33 62)(10 63 34)(11 35 64)(12 49 36)(13 37 50)(14 51 38)(15 39 52)(16 53 40)(17 68 88)(18 89 69)(19 70 90)(20 91 71)(21 72 92)(22 93 73)(23 74 94)(24 95 75)(25 76 96)(26 81 77)(27 78 82)(28 83 79)(29 80 84)(30 85 65)(31 66 86)(32 87 67)
(1 54 41)(2 55 42)(3 56 43)(4 57 44)(5 58 45)(6 59 46)(7 60 47)(8 61 48)(9 62 33)(10 63 34)(11 64 35)(12 49 36)(13 50 37)(14 51 38)(15 52 39)(16 53 40)(17 68 88)(18 69 89)(19 70 90)(20 71 91)(21 72 92)(22 73 93)(23 74 94)(24 75 95)(25 76 96)(26 77 81)(27 78 82)(28 79 83)(29 80 84)(30 65 85)(31 66 86)(32 67 87)
(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 25 9 17)(2 24 10 32)(3 23 11 31)(4 22 12 30)(5 21 13 29)(6 20 14 28)(7 19 15 27)(8 18 16 26)(33 68 41 76)(34 67 42 75)(35 66 43 74)(36 65 44 73)(37 80 45 72)(38 79 46 71)(39 78 47 70)(40 77 48 69)(49 85 57 93)(50 84 58 92)(51 83 59 91)(52 82 60 90)(53 81 61 89)(54 96 62 88)(55 95 63 87)(56 94 64 86)

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

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

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

42 conjugacy classes

class 1  2 3A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F12G16A16B16C16D24A24B24C24D24E···24J48A···48H
order123334446668812121212121212161616162424242424···2448···48
size112242247222422224442424666622224···46···6

42 irreducible representations

dim11112222222222444444
type++++++++++-+-+++-+-
imageC1C2C2C2S3S3D4D6D8D12C3⋊D4Q32D24Dic24S32D4⋊S3C3⋊D12C3⋊Q32C3⋊D24C323Q32
kernelC323Q32C3×C3⋊C16C3×Dic12C325Q16C3⋊C16Dic12C3×C12C24C3×C6C12C12C32C6C3C8C6C4C3C2C1
# reps11111112222448111224

Matrix representation of C323Q32 in GL6(𝔽97)

100000
010000
001000
000100
0000096
0000196
,
100000
010000
0009600
0019600
000010
000001
,
2440000
95280000
00293900
00586800
00008256
00004115
,
95750000
8420000
00392900
00685800
00004115
00008256

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,95,0,0,0,0,4,28,0,0,0,0,0,0,29,58,0,0,0,0,39,68,0,0,0,0,0,0,82,41,0,0,0,0,56,15],[95,84,0,0,0,0,75,2,0,0,0,0,0,0,39,68,0,0,0,0,29,58,0,0,0,0,0,0,41,82,0,0,0,0,15,56] >;

C323Q32 in GAP, Magma, Sage, TeX

C_3^2\rtimes_3Q_{32}
% in TeX

G:=Group("C3^2:3Q32");
// GroupNames label

G:=SmallGroup(288,199);
// by ID

G=gap.SmallGroup(288,199);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,254,142,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽