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G = C3317SD16order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C3317SD16, C12.54S32, (C3×C6).39D12, C32(C242S3), C324Q87S3, (C3×C12).120D6, (C32×C6).37D4, C328(C24⋊C2), C6.14(C12⋊S3), C3312D4.3C2, C2.6(C338D4), C31(C325SD16), C6.10(C3⋊D12), C3211(Q82S3), (C32×C12).16C22, (C3×C3⋊C8)⋊3S3, C3⋊C83(C3⋊S3), C4.3(S3×C3⋊S3), (C32×C3⋊C8)⋊4C2, C12.14(C2×C3⋊S3), (C3×C324Q8)⋊3C2, (C3×C6).79(C3⋊D4), SmallGroup(432,444)

Series: Derived Chief Lower central Upper central

C1C32×C12 — C3317SD16
C1C3C32C33C32×C6C32×C12C32×C3⋊C8 — C3317SD16
C33C32×C6C32×C12 — C3317SD16
C1C2C4

Generators and relations for C3317SD16
 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], Q82S3, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C3×C24, C3×Dic6 [×4], C324Q8, C12⋊S3 [×9], C3×C3⋊Dic3, C32×C12, C2×C33⋊C2, C325SD16 [×4], C242S3, C32×C3⋊C8, C3×C324Q8, C3312D4, C3317SD16
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], Q82S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C325SD16 [×4], C242S3, C338D4, C3317SD16

Smallest permutation representation of C3317SD16
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⋊C2S32Q82S3C3⋊D12C325SD16
kernelC3317SD16C32×C3⋊C8C3×C324Q8C3312D4C3×C3⋊C8C324Q8C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps11114115282164148

Matrix representation of C3317SD16 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] >;

C3317SD16 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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