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G = C323SD32order 288 = 25·32

2nd semidirect product of C32 and SD32 acting via SD32/C8=C22

metabelian, supersoluble, monomial

Aliases: D24.1S3, C6.13D24, C24.16D6, C323SD32, C12.10D12, C8.5S32, C3⋊C162S3, (C3×C6).9D8, C6.2(D4⋊S3), C31(D8.S3), (C3×D24).3C2, C32(C48⋊C2), (C3×C12).24D4, C325Q164C2, (C3×C24).9C22, C4.2(C3⋊D12), C2.5(C3⋊D24), C12.67(C3⋊D4), (C3×C3⋊C16)⋊2C2, SmallGroup(288,196)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C323SD32
C1C3C32C3×C6C3×C12C3×C24C3×D24 — C323SD32
C32C3×C6C3×C12C3×C24 — C323SD32
C1C2C4C8

Generators and relations for C323SD32
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd=b-1, dcd=c7 >

Subgroups: 322 in 61 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C32, Dic3 [×4], C12 [×2], C12, D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24 [×2], C24, Dic6 [×4], D12, C3×D4, SD32, C3⋊Dic3, C3×C12, S3×C6, C3⋊C16, C48, D24, Dic12 [×3], C3×D8, C3×C24, C3×D12, C324Q8, C48⋊C2, D8.S3, C3×C3⋊C16, C3×D24, C325Q16, C323SD32
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, D12, C3⋊D4, SD32, S32, D24, D4⋊S3, C3⋊D12, C48⋊C2, D8.S3, C3⋊D24, C323SD32

Smallest permutation representation of C323SD32
On 96 points
Generators in S96
(1 47 30)(2 31 48)(3 33 32)(4 17 34)(5 35 18)(6 19 36)(7 37 20)(8 21 38)(9 39 22)(10 23 40)(11 41 24)(12 25 42)(13 43 26)(14 27 44)(15 45 28)(16 29 46)(49 96 65)(50 66 81)(51 82 67)(52 68 83)(53 84 69)(54 70 85)(55 86 71)(56 72 87)(57 88 73)(58 74 89)(59 90 75)(60 76 91)(61 92 77)(62 78 93)(63 94 79)(64 80 95)
(1 30 47)(2 31 48)(3 32 33)(4 17 34)(5 18 35)(6 19 36)(7 20 37)(8 21 38)(9 22 39)(10 23 40)(11 24 41)(12 25 42)(13 26 43)(14 27 44)(15 28 45)(16 29 46)(49 65 96)(50 66 81)(51 67 82)(52 68 83)(53 69 84)(54 70 85)(55 71 86)(56 72 87)(57 73 88)(58 74 89)(59 75 90)(60 76 91)(61 77 92)(62 78 93)(63 79 94)(64 80 95)
(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 78)(2 69)(3 76)(4 67)(5 74)(6 65)(7 72)(8 79)(9 70)(10 77)(11 68)(12 75)(13 66)(14 73)(15 80)(16 71)(17 51)(18 58)(19 49)(20 56)(21 63)(22 54)(23 61)(24 52)(25 59)(26 50)(27 57)(28 64)(29 55)(30 62)(31 53)(32 60)(33 91)(34 82)(35 89)(36 96)(37 87)(38 94)(39 85)(40 92)(41 83)(42 90)(43 81)(44 88)(45 95)(46 86)(47 93)(48 84)

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

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

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

42 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C6D6E8A8B12A12B12C12D12E16A16B16C16D24A24B24C24D24E···24J48A···48H
order1223334466666881212121212161616162424242424···2448···48
size112422427222424242222444666622224···46···6

42 irreducible representations

dim11112222222222444444
type++++++++++++++-+-
imageC1C2C2C2S3S3D4D6D8D12C3⋊D4SD32D24C48⋊C2S32D4⋊S3C3⋊D12D8.S3C3⋊D24C323SD32
kernelC323SD32C3×C3⋊C16C3×D24C325Q16C3⋊C16D24C3×C12C24C3×C6C12C12C32C6C3C8C6C4C3C2C1
# reps11111112222448111224

Matrix representation of C323SD32 in GL4(𝔽97) generated by

1000
0100
00096
00196
,
09600
19600
0010
0001
,
56700
307200
0001
0010
,
167900
958100
0010
0001
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,96,96],[0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[5,30,0,0,67,72,0,0,0,0,0,1,0,0,1,0],[16,95,0,0,79,81,0,0,0,0,1,0,0,0,0,1] >;

C323SD32 in GAP, Magma, Sage, TeX

C_3^2\rtimes_3{\rm SD}_{32}
% in TeX

G:=Group("C3^2:3SD32");
// GroupNames label

G:=SmallGroup(288,196);
// by ID

G=gap.SmallGroup(288,196);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,1356,9414]);
// 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,b*c=c*b,d*b*d=b^-1,d*c*d=c^7>;
// generators/relations

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