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## G = C2×C32⋊5SD16order 288 = 25·32

### Direct product of C2 and C32⋊5SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C32⋊5SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊5SD16 — C2×C32⋊5SD16
 Lower central C32 — C3×C6 — C3×C12 — C2×C32⋊5SD16
 Upper central C1 — C22 — C2×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 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 36 36 2 2 4 2 2 12 12 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 SD16 D12 C3⋊D4 D12 C3⋊D4 C24⋊C2 S32 Q8⋊2S3 C3⋊D12 C2×S32 C3⋊D12 C32⋊5SD16 kernel C2×C32⋊5SD16 C32⋊5SD16 C6×C3⋊C8 C6×Dic6 C2×C12⋊S3 C2×C3⋊C8 C2×Dic6 C3×C12 C62 C3⋊C8 Dic6 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 1 1 1 4

Matrix representation of C2×C325SD16 in GL6(𝔽73)

 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 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 6 0 0 0 0 67 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

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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