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G = C2×C325SD16order 288 = 25·32

Direct product of C2 and C325SD16

direct product, metabelian, supersoluble, monomial

Aliases: C2×C325SD16, Dic617D6, C12.24D12, C62.51D4, C3⋊C825D6, (C3×C6)⋊5SD16, C62(C24⋊C2), (C2×Dic6)⋊1S3, (C6×Dic6)⋊3C2, (C2×C6).61D12, (C3×C12).69D4, C6.71(C2×D12), C328(C2×SD16), C61(Q82S3), (C2×C12).121D6, C4.7(C3⋊D12), C12.72(C3⋊D4), (C6×C12).81C22, (C3×C12).68C23, C12.127(C22×S3), (C3×Dic6)⋊21C22, C12⋊S3.27C22, C22.21(C3⋊D12), (C2×C3⋊C8)⋊8S3, (C6×C3⋊C8)⋊12C2, C4.56(C2×S32), (C2×C4).67S32, C33(C2×C24⋊C2), C6.7(C2×C3⋊D4), (C3×C3⋊C8)⋊30C22, C31(C2×Q82S3), (C3×C6).72(C2×D4), C2.11(C2×C3⋊D12), (C2×C6).39(C3⋊D4), (C2×C12⋊S3).11C2, SmallGroup(288,480)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C2×C325SD16
C1C3C32C3×C6C3×C12C3×Dic6C325SD16 — C2×C325SD16
C32C3×C6C3×C12 — C2×C325SD16
C1C22C2×C4

Generators and relations for C2×C325SD16
 G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d3 >

Subgroups: 834 in 163 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×16], C2×C6 [×2], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, D12 [×10], C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×3], C22×S3 [×4], C2×SD16, C3×Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×4], C62, C24⋊C2 [×4], C2×C3⋊C8, Q82S3 [×4], C2×C24, C2×Dic6, C2×D12 [×3], C6×Q8, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C12⋊S3 [×2], C12⋊S3, C6×C12, C22×C3⋊S3, C2×C24⋊C2, C2×Q82S3, C325SD16 [×4], C6×C3⋊C8, C6×Dic6, C2×C12⋊S3, C2×C325SD16
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C2×SD16, S32, C24⋊C2 [×2], Q82S3 [×2], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C2×C24⋊C2, C2×Q82S3, C325SD16 [×2], C2×C3⋊D12, C2×C325SD16

Smallest permutation representation of C2×C325SD16
On 48 points
Generators in S48
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 21 11)(2 12 22)(3 23 13)(4 14 24)(5 17 15)(6 16 18)(7 19 9)(8 10 20)(25 45 33)(26 34 46)(27 47 35)(28 36 48)(29 41 37)(30 38 42)(31 43 39)(32 40 44)
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 17)(6 16 18)(7 9 19)(8 10 20)(25 33 45)(26 34 46)(27 35 47)(28 36 48)(29 37 41)(30 38 42)(31 39 43)(32 40 44)
(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)
(1 3)(2 6)(5 7)(9 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)(25 47)(26 42)(27 45)(28 48)(29 43)(30 46)(31 41)(32 44)(33 35)(34 38)(37 39)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6F6G6H6I8A8B8C8D12A12B12C12D12E···12J12K12L12M12N24A···24H
order12222233344446···666688881212121212···121212121224···24
size111136362242212122···2444666622224···4121212126···6

48 irreducible representations

dim111112222222222222444444
type++++++++++++++++++++
imageC1C2C2C2C2S3S3D4D4D6D6D6SD16D12C3⋊D4D12C3⋊D4C24⋊C2S32Q82S3C3⋊D12C2×S32C3⋊D12C325SD16
kernelC2×C325SD16C325SD16C6×C3⋊C8C6×Dic6C2×C12⋊S3C2×C3⋊C8C2×Dic6C3×C12C62C3⋊C8Dic6C2×C12C3×C6C12C12C2×C6C2×C6C6C2×C4C6C4C4C22C2
# reps141111111222422228121114

Matrix representation of C2×C325SD16 in GL6(𝔽73)

7200000
0720000
0072000
0007200
000010
000001
,
100000
010000
001000
000100
0000721
0000720
,
100000
010000
0072100
0072000
000010
000001
,
660000
6760000
0072000
0007200
000001
000010
,
010000
100000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C325SD16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C2xC3^2:5SD16");
// GroupNames label

G:=SmallGroup(288,480);
// by ID

G=gap.SmallGroup(288,480);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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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