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G = C33⋊17SD16  order 432 = 24·33

9th semidirect product of C33 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊17SD16, C12.54S32, (C3×C6).39D12, C3⋊2(C24⋊2S3), C32⋊4Q8⋊7S3, (C3×C12).120D6, (C32×C6).37D4, C32⋊8(C24⋊C2), C6.14(C12⋊S3), C33⋊12D4.3C2, C2.6(C33⋊8D4), C3⋊1(C32⋊5SD16), C6.10(C3⋊D12), C32⋊11(Q8⋊2S3), (C32×C12).16C22, (C3×C3⋊C8)⋊3S3, C3⋊C8⋊3(C3⋊S3), C4.3(S3×C3⋊S3), (C32×C3⋊C8)⋊4C2, C12.14(C2×C3⋊S3), (C3×C32⋊4Q8)⋊3C2, (C3×C6).79(C3⋊D4), SmallGroup(432,444)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊17SD16
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C33⋊17SD16
Lower central
C33 — C32×C6 — C32×C12 — C33⋊17SD16
Upper central
C1 — C2 — C4

Generators and relations for C33⋊17SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d3 >

Subgroups: 1616 in 184 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3 [×4], C3 [×4], C4, C4, C22, S3 [×13], C6, C6 [×4], C6 [×4], C8, D4, Q8, C32, C32 [×4], C32 [×4], Dic3 [×4], C12, C12 [×4], C12 [×5], D6 [×13], SD16, C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C24 [×4], Dic6 [×4], D12 [×9], C3×Q8, C33, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C3×C12 [×4], C3×C12 [×4], C2×C3⋊S3 [×13], C24⋊C2 [×4], Q8⋊2S3, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C3×C24, C3×Dic6 [×4], C32⋊4Q8, C12⋊S3 [×9], C3×C3⋊Dic3, C32×C12, C2×C33⋊C2, C32⋊5SD16 [×4], C24⋊2S3, C32×C3⋊C8, C3×C32⋊4Q8, C33⋊12D4, C33⋊17SD16
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], SD16, C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, C24⋊C2 [×4], Q8⋊2S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C32⋊5SD16 [×4], C24⋊2S3, C33⋊8D4, C33⋊17SD16

Smallest permutation representation of C33⋊17SD16
►On 72 points
Generators in S72
(1 27 44)(2 28 45)(3 29 46)(4 30 47)(5 31 48)(6 32 41)(7 25 42)(8 26 43)(9 63 72)(10 64 65)(11 57 66)(12 58 67)(13 59 68)(14 60 69)(15 61 70)(16 62 71)(17 53 39)(18 54 40)(19 55 33)(20 56 34)(21 49 35)(22 50 36)(23 51 37)(24 52 38)

(1 22 66)(2 23 67)(3 24 68)(4 17 69)(5 18 70)(6 19 71)(7 20 72)(8 21 65)(9 25 56)(10 26 49)(11 27 50)(12 28 51)(13 29 52)(14 30 53)(15 31 54)(16 32 55)(33 62 41)(34 63 42)(35 64 43)(36 57 44)(37 58 45)(38 59 46)(39 60 47)(40 61 48)

(1 36 11)(2 12 37)(3 38 13)(4 14 39)(5 40 15)(6 16 33)(7 34 9)(8 10 35)(17 30 60)(18 61 31)(19 32 62)(20 63 25)(21 26 64)(22 57 27)(23 28 58)(24 59 29)(41 71 55)(42 56 72)(43 65 49)(44 50 66)(45 67 51)(46 52 68)(47 69 53)(48 54 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)

(1 5)(2 8)(4 6)(9 34)(10 37)(11 40)(12 35)(13 38)(14 33)(15 36)(16 39)(17 71)(18 66)(19 69)(20 72)(21 67)(22 70)(23 65)(24 68)(25 42)(26 45)(27 48)(28 43)(29 46)(30 41)(31 44)(32 47)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)
 
G:=sub<Sym(72)| (1,27,44)(2,28,45)(3,29,46)(4,30,47)(5,31,48)(6,32,41)(7,25,42)(8,26,43)(9,63,72)(10,64,65)(11,57,66)(12,58,67)(13,59,68)(14,60,69)(15,61,70)(16,62,71)(17,53,39)(18,54,40)(19,55,33)(20,56,34)(21,49,35)(22,50,36)(23,51,37)(24,52,38), (1,22,66)(2,23,67)(3,24,68)(4,17,69)(5,18,70)(6,19,71)(7,20,72)(8,21,65)(9,25,56)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(33,62,41)(34,63,42)(35,64,43)(36,57,44)(37,58,45)(38,59,46)(39,60,47)(40,61,48), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,30,60)(18,61,31)(19,32,62)(20,63,25)(21,26,64)(22,57,27)(23,28,58)(24,59,29)(41,71,55)(42,56,72)(43,65,49)(44,50,66)(45,67,51)(46,52,68)(47,69,53)(48,54,70), (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), (1,5)(2,8)(4,6)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,71)(18,66)(19,69)(20,72)(21,67)(22,70)(23,65)(24,68)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63)>;
 
G:=Group( (1,27,44)(2,28,45)(3,29,46)(4,30,47)(5,31,48)(6,32,41)(7,25,42)(8,26,43)(9,63,72)(10,64,65)(11,57,66)(12,58,67)(13,59,68)(14,60,69)(15,61,70)(16,62,71)(17,53,39)(18,54,40)(19,55,33)(20,56,34)(21,49,35)(22,50,36)(23,51,37)(24,52,38), (1,22,66)(2,23,67)(3,24,68)(4,17,69)(5,18,70)(6,19,71)(7,20,72)(8,21,65)(9,25,56)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(33,62,41)(34,63,42)(35,64,43)(36,57,44)(37,58,45)(38,59,46)(39,60,47)(40,61,48), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,30,60)(18,61,31)(19,32,62)(20,63,25)(21,26,64)(22,57,27)(23,28,58)(24,59,29)(41,71,55)(42,56,72)(43,65,49)(44,50,66)(45,67,51)(46,52,68)(47,69,53)(48,54,70), (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), (1,5)(2,8)(4,6)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,71)(18,66)(19,69)(20,72)(21,67)(22,70)(23,65)(24,68)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(49,58)(50,61)(51,64)(52,59)(53,62)(54,57)(55,60)(56,63) );
 
G=PermutationGroup([(1,27,44),(2,28,45),(3,29,46),(4,30,47),(5,31,48),(6,32,41),(7,25,42),(8,26,43),(9,63,72),(10,64,65),(11,57,66),(12,58,67),(13,59,68),(14,60,69),(15,61,70),(16,62,71),(17,53,39),(18,54,40),(19,55,33),(20,56,34),(21,49,35),(22,50,36),(23,51,37),(24,52,38)], [(1,22,66),(2,23,67),(3,24,68),(4,17,69),(5,18,70),(6,19,71),(7,20,72),(8,21,65),(9,25,56),(10,26,49),(11,27,50),(12,28,51),(13,29,52),(14,30,53),(15,31,54),(16,32,55),(33,62,41),(34,63,42),(35,64,43),(36,57,44),(37,58,45),(38,59,46),(39,60,47),(40,61,48)], [(1,36,11),(2,12,37),(3,38,13),(4,14,39),(5,40,15),(6,16,33),(7,34,9),(8,10,35),(17,30,60),(18,61,31),(19,32,62),(20,63,25),(21,26,64),(22,57,27),(23,28,58),(24,59,29),(41,71,55),(42,56,72),(43,65,49),(44,50,66),(45,67,51),(46,52,68),(47,69,53),(48,54,70)], [(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)], [(1,5),(2,8),(4,6),(9,34),(10,37),(11,40),(12,35),(13,38),(14,33),(15,36),(16,39),(17,71),(18,66),(19,69),(20,72),(21,67),(22,70),(23,65),(24,68),(25,42),(26,45),(27,48),(28,43),(29,46),(30,41),(31,44),(32,47),(49,58),(50,61),(51,64),(52,59),(53,62),(54,57),(55,60),(56,63)])
 

60 conjugacy classes

class 1 2A2B3A···3E3F3G3H3I4A4B6A···6E6F6G6H6I8A8B12A···12H12I···12Q12R12S24A···24P
order1223···33333446···666668812···1212···12121224···24
size111082···244442362···24444662···24···436366···6

60 irreducible representations

dim1111222222224444
type+++++++++++++
imageC1C2C2C2S3S3D4D6SD16D12C3⋊D4C24⋊C2S32Q8⋊2S3C3⋊D12C32⋊5SD16
kernelC33⋊17SD16C32×C3⋊C8C3×C32⋊4Q8C33⋊12D4C3×C3⋊C8C32⋊4Q8C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps11114115282164148

Matrix representation of C33⋊17SD16 ►in GL8(𝔽73)

721000000
720000000
00100000
00010000
000007200
000017200
00000010
00000001
,
721000000
720000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000072
000000172
,
720000000
072000000
000550000
004610000
000072000
000007200
00000001
00000010
,
01000000
10000000
007200000
004810000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(73))| [72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,55,61,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,48,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C33⋊17SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_{17}{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:17SD16");
 
// GroupNames label
 
G:=SmallGroup(432,444);
 
// by ID
 
G=gap.SmallGroup(432,444);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,197,64,135,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

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