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## G = C32⋊3SD32order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C32⋊3SD32
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×D24 — C32⋊3SD32
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C32⋊3SD32
 Upper central C1 — C2 — C4 — C8

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, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24, C24, Dic6, D12, C3×D4, SD32, C3⋊Dic3, C3×C12, S3×C6, C3⋊C16, C48, D24, Dic12, C3×D8, C3×C24, C3×D12, C324Q8, C48⋊C2, D8.S3, C3×C3⋊C16, C3×D24, C325Q16, C323SD32
Quotients: C1, C2, C22, S3, D4, D6, 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 38 68)(2 69 39)(3 40 70)(4 71 41)(5 42 72)(6 73 43)(7 44 74)(8 75 45)(9 46 76)(10 77 47)(11 48 78)(12 79 33)(13 34 80)(14 65 35)(15 36 66)(16 67 37)(17 54 82)(18 83 55)(19 56 84)(20 85 57)(21 58 86)(22 87 59)(23 60 88)(24 89 61)(25 62 90)(26 91 63)(27 64 92)(28 93 49)(29 50 94)(30 95 51)(31 52 96)(32 81 53)
(1 68 38)(2 69 39)(3 70 40)(4 71 41)(5 72 42)(6 73 43)(7 74 44)(8 75 45)(9 76 46)(10 77 47)(11 78 48)(12 79 33)(13 80 34)(14 65 35)(15 66 36)(16 67 37)(17 54 82)(18 55 83)(19 56 84)(20 57 85)(21 58 86)(22 59 87)(23 60 88)(24 61 89)(25 62 90)(26 63 91)(27 64 92)(28 49 93)(29 50 94)(30 51 95)(31 52 96)(32 53 81)
(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 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 61)(34 52)(35 59)(36 50)(37 57)(38 64)(39 55)(40 62)(41 53)(42 60)(43 51)(44 58)(45 49)(46 56)(47 63)(48 54)(65 87)(66 94)(67 85)(68 92)(69 83)(70 90)(71 81)(72 88)(73 95)(74 86)(75 93)(76 84)(77 91)(78 82)(79 89)(80 96)

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 3 3 3 4 4 6 6 6 6 6 8 8 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 24 2 2 4 2 72 2 2 4 24 24 2 2 2 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 D8 D12 C3⋊D4 SD32 D24 C48⋊C2 S32 D4⋊S3 C3⋊D12 D8.S3 C3⋊D24 C32⋊3SD32 kernel C32⋊3SD32 C3×C3⋊C16 C3×D24 C32⋊5Q16 C3⋊C16 D24 C3×C12 C24 C3×C6 C12 C12 C32 C6 C3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 2 4 4 8 1 1 1 2 2 4

Matrix representation of C323SD32 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 0 96 0 0 1 96
,
 0 96 0 0 1 96 0 0 0 0 1 0 0 0 0 1
,
 5 67 0 0 30 72 0 0 0 0 0 1 0 0 1 0
,
 16 79 0 0 95 81 0 0 0 0 1 0 0 0 0 1
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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