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G = C3315SD16order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C3315SD16, C12.30S32, (C3×Dic6)⋊3S3, (C3×C6).80D12, Dic61(C3⋊S3), C324C810S3, (C3×C12).118D6, (C32×C6).35D4, (C32×Dic6)⋊4C2, C3312D4.2C2, C6.3(C327D4), C2.6(C337D4), C3213(C24⋊C2), C32(C325SD16), C6.24(C3⋊D12), C31(C3211SD16), C328(Q82S3), (C32×C12).14C22, C4.10(S3×C3⋊S3), C12.32(C2×C3⋊S3), (C3×C324C8)⋊5C2, (C3×C6).58(C3⋊D4), SmallGroup(432,442)

Series: Derived Chief Lower central Upper central

C1C32×C12 — C3315SD16
C1C3C32C33C32×C6C32×C12C32×Dic6 — C3315SD16
C33C32×C6C32×C12 — C3315SD16
C1C2C4

Generators and relations for C3315SD16
 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, Q82S3 [×4], C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C324C8, C3×Dic6 [×4], C12⋊S3 [×9], Q8×C32, C32×Dic3, C32×C12, C2×C33⋊C2, C325SD16 [×4], C3211SD16, C3×C324C8, C32×Dic6, C3312D4, C3315SD16
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], SD16, C3⋊S3, D12, C3⋊D4 [×4], S32 [×4], C2×C3⋊S3, C24⋊C2, Q82S3 [×4], C3⋊D12 [×4], C327D4, S3×C3⋊S3, C325SD16 [×4], C3211SD16, C337D4, C3315SD16

Smallest permutation representation of C3315SD16
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⋊C2S32Q82S3C3⋊D12C325SD16
kernelC3315SD16C3×C324C8C32×Dic6C3312D4C324C8C3×Dic6C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C3
# reps1111141522844448

Matrix representation of C3315SD16 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] >;

C3315SD16 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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