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G = C322D16order 288 = 25·32

1st semidirect product of C32 and D16 acting via D16/C8=C22

metabelian, supersoluble, monomial

Aliases: D243S3, C322D16, C24.14D6, C8.13S32, (C3×C6).6D8, (C3×D24)⋊5C2, C32(C3⋊D16), C6.8(D4⋊S3), (C3×C12).21D4, C24.S31C2, (C3×C24).6C22, C4.1(D6⋊S3), C12.21(C3⋊D4), C2.3(C322D8), SmallGroup(288,193)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C322D16
C1C3C32C3×C6C3×C12C3×C24C3×D24 — C322D16
C32C3×C6C3×C12C3×C24 — C322D16
C1C2C4C8

Generators and relations for C322D16
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 302 in 61 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, D4 [×2], C32, C12 [×2], C12, D6 [×2], C2×C6 [×2], C16, D8 [×2], C3×S3 [×2], C3×C6, C24 [×2], C24, D12 [×2], C3×D4 [×2], D16, C3×C12, S3×C6 [×2], C3⋊C16 [×3], D24 [×2], C3×D8 [×2], C3×C24, C3×D12 [×2], C3⋊D16 [×2], C24.S3, C3×D24 [×2], C322D16
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, C3⋊D4 [×2], D16, S32, D4⋊S3 [×2], D6⋊S3, C3⋊D16 [×2], C322D8, C322D16

Smallest permutation representation of C322D16
On 96 points
Generators in S96
(1 41 32)(2 17 42)(3 43 18)(4 19 44)(5 45 20)(6 21 46)(7 47 22)(8 23 48)(9 33 24)(10 25 34)(11 35 26)(12 27 36)(13 37 28)(14 29 38)(15 39 30)(16 31 40)(49 71 93)(50 94 72)(51 73 95)(52 96 74)(53 75 81)(54 82 76)(55 77 83)(56 84 78)(57 79 85)(58 86 80)(59 65 87)(60 88 66)(61 67 89)(62 90 68)(63 69 91)(64 92 70)
(1 32 41)(2 42 17)(3 18 43)(4 44 19)(5 20 45)(6 46 21)(7 22 47)(8 48 23)(9 24 33)(10 34 25)(11 26 35)(12 36 27)(13 28 37)(14 38 29)(15 30 39)(16 40 31)(49 71 93)(50 94 72)(51 73 95)(52 96 74)(53 75 81)(54 82 76)(55 77 83)(56 84 78)(57 79 85)(58 86 80)(59 65 87)(60 88 66)(61 67 89)(62 90 68)(63 69 91)(64 92 70)
(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 50)(2 49)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 86)(34 85)(35 84)(36 83)(37 82)(38 81)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E6F6G8A8B12A12B12C12D16A16B16C16D24A···24H
order12223334666666688121212121616161624···24
size112424224222424242424224444181818184···4

33 irreducible representations

dim111222222444444
type++++++++++-+
imageC1C2C2S3D4D6D8C3⋊D4D16S32D4⋊S3D6⋊S3C3⋊D16C322D8C322D16
kernelC322D16C24.S3C3×D24D24C3×C12C24C3×C6C12C32C8C6C4C3C2C1
# reps112212244121424

Matrix representation of C322D16 in GL6(𝔽97)

100000
010000
001000
000100
0000096
0000196
,
100000
010000
0009600
0019600
000010
000001
,
26950000
2260000
000100
001000
000001
000010
,
79680000
68180000
0009600
0096000
000010
000001

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],[26,2,0,0,0,0,95,26,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[79,68,0,0,0,0,68,18,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C322D16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2D_{16}
% in TeX

G:=Group("C3^2:2D16");
// GroupNames label

G:=SmallGroup(288,193);
// by ID

G=gap.SmallGroup(288,193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,1356,9414]);
// Polycyclic

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

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