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G = C3×S3×SD16order 288 = 25·32

Direct product of C3, S3 and SD16

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×SD16, C2421D6, C85(S3×C6), (S3×D4).C6, (S3×C8)⋊4C6, C245(C2×C6), Q83(S3×C6), (S3×Q8)⋊4C6, C24⋊C25C6, D4.S33C6, (C3×Q8)⋊14D6, D4.2(S3×C6), C32(C6×SD16), C6.30(C6×D4), (S3×C24)⋊10C2, Q82S31C6, (C3×SD16)⋊3C6, Dic62(C2×C6), (C3×D4).25D6, D12.2(C2×C6), D6.13(C3×D4), (S3×C6).49D4, C6.190(S3×D4), (C3×C24)⋊16C22, C12.4(C22×C6), Dic3.4(C3×D4), C3215(C2×SD16), (C3×C12).75C23, (C3×Dic3).31D4, (C32×SD16)⋊5C2, (Q8×C32)⋊5C22, (S3×C12).49C22, C12.155(C22×S3), (C3×Dic6)⋊12C22, (C3×D12).26C22, (D4×C32).12C22, C3⋊C86(C2×C6), C4.4(S3×C2×C6), (C3×S3×Q8)⋊4C2, (C3×S3×D4).2C2, C2.18(C3×S3×D4), (C3×Q8)⋊2(C2×C6), (C3×C3⋊C8)⋊38C22, (C4×S3).9(C2×C6), (C3×D4).2(C2×C6), (C3×C24⋊C2)⋊13C2, (C3×D4.S3)⋊14C2, (C3×C6).218(C2×D4), (C3×Q82S3)⋊13C2, SmallGroup(288,684)

Series: Derived Chief Lower central Upper central

C1C12 — C3×S3×SD16
C1C3C6C12C3×C12S3×C12C3×S3×D4 — C3×S3×SD16
C3C6C12 — C3×S3×SD16
C1C6C12C3×SD16

Generators and relations for C3×S3×SD16
 G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 402 in 146 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3, C6 [×2], C6 [×7], C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6 [×3], C2×C6 [×7], C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×D4 [×3], C3×Q8 [×2], C3×Q8 [×3], C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6 [×3], C62, S3×C8, C24⋊C2, D4.S3, Q82S3, C2×C24, C3×SD16 [×2], C3×SD16 [×4], S3×D4, S3×Q8, C6×D4, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, S3×SD16, C6×SD16, S3×C24, C3×C24⋊C2, C3×D4.S3, C3×Q82S3, C32×SD16, C3×S3×D4, C3×S3×Q8, C3×S3×SD16
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], SD16 [×2], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C2×SD16, S3×C6 [×3], C3×SD16 [×2], S3×D4, C6×D4, S3×C2×C6, S3×SD16, C6×SD16, C3×S3×D4, C3×S3×SD16

Smallest permutation representation of C3×S3×SD16
On 48 points
Generators in S48
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 17 42)(10 18 43)(11 19 44)(12 20 45)(13 21 46)(14 22 47)(15 23 48)(16 24 41)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)

G:=sub<Sym(48)| (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,17,42)(10,18,43)(11,19,44)(12,20,45)(13,21,46)(14,22,47)(15,23,48)(16,24,41), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)>;

G:=Group( (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,17,42)(10,18,43)(11,19,44)(12,20,45)(13,21,46)(14,22,47)(15,23,48)(16,24,41), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48) );

G=PermutationGroup([(1,25,40),(2,26,33),(3,27,34),(4,28,35),(5,29,36),(6,30,37),(7,31,38),(8,32,39),(9,42,17),(10,43,18),(11,44,19),(12,45,20),(13,46,21),(14,47,22),(15,48,23),(16,41,24)], [(1,25,40),(2,26,33),(3,27,34),(4,28,35),(5,29,36),(6,30,37),(7,31,38),(8,32,39),(9,17,42),(10,18,43),(11,19,44),(12,20,45),(13,21,46),(14,22,47),(15,23,48),(16,24,41)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48)])

63 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P8A8B8C8D12A12B12C···12G12H12I12J12K12L12M12N24A24B24C24D24E···24J24K24L24M24N
order12222233333444466666666666666668888121212···12121212121212122424242424···2424242424
size113341211222246121122233334488812122266224···466888121222224···46666

63 irreducible representations

dim1111111111111111222222222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6S3D4D4D6D6D6SD16C3×S3C3×D4C3×D4S3×C6S3×C6S3×C6C3×SD16S3×D4S3×SD16C3×S3×D4C3×S3×SD16
kernelC3×S3×SD16S3×C24C3×C24⋊C2C3×D4.S3C3×Q82S3C32×SD16C3×S3×D4C3×S3×Q8S3×SD16S3×C8C24⋊C2D4.S3Q82S3C3×SD16S3×D4S3×Q8C3×SD16C3×Dic3S3×C6C24C3×D4C3×Q8C3×S3SD16Dic3D6C8D4Q8S3C6C3C2C1
# reps1111111122222222111111422222281224

Matrix representation of C3×S3×SD16 in GL4(𝔽73) generated by

64000
06400
0010
0001
,
64000
65800
0010
0001
,
1700
07200
0010
0001
,
72000
07200
00061
00612
,
72000
07200
0010
007272
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,65,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,7,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,6,0,0,61,12],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

C3×S3×SD16 in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm SD}_{16}
% in TeX

G:=Group("C3xS3xSD16");
// GroupNames label

G:=SmallGroup(288,684);
// by ID

G=gap.SmallGroup(288,684);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,268,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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