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G = C33⋊15SD16  order 432 = 24·33

7th semidirect product of C33 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊15SD16, C12.30S32, (C3×Dic6)⋊3S3, (C3×C6).80D12, Dic6⋊1(C3⋊S3), C32⋊4C8⋊10S3, (C3×C12).118D6, (C32×C6).35D4, (C32×Dic6)⋊4C2, C33⋊12D4.2C2, C6.3(C32⋊7D4), C2.6(C33⋊7D4), C32⋊13(C24⋊C2), C3⋊2(C32⋊5SD16), C6.24(C3⋊D12), C3⋊1(C32⋊11SD16), C32⋊8(Q8⋊2S3), (C32×C12).14C22, C4.10(S3×C3⋊S3), C12.32(C2×C3⋊S3), (C3×C32⋊4C8)⋊5C2, (C3×C6).58(C3⋊D4), SmallGroup(432,442)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊15SD16
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×Dic6 — C33⋊15SD16
Lower central
C33 — C32×C6 — C32×C12 — C33⋊15SD16
Upper central
C1 — C2 — C4

Generators and relations for C33⋊15SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d3 >

Subgroups: 1576 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, C12, C12 [×4], C12 [×8], D6 [×13], SD16, C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8 [×4], C24, Dic6, D12 [×9], C3×Q8 [×4], C33, C3×Dic3 [×4], C3×C12, C3×C12 [×4], C3×C12 [×5], C2×C3⋊S3 [×13], C24⋊C2, Q8⋊2S3 [×4], C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C32⋊4C8, C3×Dic6 [×4], C12⋊S3 [×9], Q8×C32, C32×Dic3, C32×C12, C2×C33⋊C2, C32⋊5SD16 [×4], C32⋊11SD16, C3×C32⋊4C8, C32×Dic6, C33⋊12D4, C33⋊15SD16
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], SD16, C3⋊S3, D12, C3⋊D4 [×4], S32 [×4], C2×C3⋊S3, C24⋊C2, Q8⋊2S3 [×4], C3⋊D12 [×4], C32⋊7D4, S3×C3⋊S3, C32⋊5SD16 [×4], C32⋊11SD16, C33⋊7D4, C33⋊15SD16

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

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

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

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

51 conjugacy classes

class 1 2A2B3A···3E3F3G3H3I4A4B6A···6E6F6G6H6I8A8B12A12B12C···12N12O···12V24A24B24C24D
order1223···33333446···6666688121212···1212···1224242424
size111082···244442122···244441818224···412···1218181818

51 irreducible representations

dim1111222222224444
type+++++++++++++
imageC1C2C2C2S3S3D4D6SD16D12C3⋊D4C24⋊C2S32Q8⋊2S3C3⋊D12C32⋊5SD16
kernelC33⋊15SD16C3×C32⋊4C8C32×Dic6C33⋊12D4C32⋊4C8C3×Dic6C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps1111141522844448

Matrix representation of C33⋊15SD16 ►in GL8(𝔽73)

01000000
7272000000
00100000
00010000
000007200
000017200
00000010
00000001
,
7272000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000007272
00000010
,
10000000
7272000000
0061120000
006700000
000007200
000072000
00000010
00000001
,
10000000
7272000000
001710000
000720000
00000100
00001000
00000010
0000007272

G:=sub<GL(8,GF(73))| [0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,61,67,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;
 

C33⋊15SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_{15}{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:15SD16");
 
// GroupNames label
 
G:=SmallGroup(432,442);
 
// by ID
 
G=gap.SmallGroup(432,442);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,2028,14118]);
 
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

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