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## G = C2×C32⋊2SD16order 288 = 25·32

### Direct product of C2 and C32⋊2SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C2×C32⋊2SD16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊2SD16 — C2×C32⋊2SD16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C2×C32⋊2SD16
 Upper central C1 — C22

Generators and relations for C2×C322SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c-1, ebe=b-1, dcd-1=b, ce=ec, ede=d3 >

Subgroups: 560 in 114 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×4], C22, C22 [×4], S3 [×2], C6 [×8], C8 [×2], C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3 [×6], C12 [×2], D6 [×4], C2×C6 [×6], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×4], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C22×S3, C22×C6, C2×SD16, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×4], C62, C2×Dic6, C2×C3⋊D4, C322C8 [×2], D6⋊S3 [×2], D6⋊S3, C322Q8 [×2], C322Q8, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C322SD16 [×4], C2×C322C8, C2×D6⋊S3, C2×C322Q8, C2×C322SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C2×SD16, S3≀C2, C322SD16 [×2], C2×S3≀C2, C2×C322SD16

Character table of C2×C322SD16

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 12 12 4 4 12 12 18 18 4 4 4 4 4 4 12 12 12 12 18 18 18 18 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 -2 -2 0 0 2 2 0 0 2 -2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 0 0 -2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 2 0 0 0 0 2 -2 -2 -2 -2 2 0 0 0 0 -√-2 √-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ12 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 2 2 -2 -2 -2 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ13 2 -2 -2 2 0 0 2 2 0 0 0 0 2 -2 -2 -2 -2 2 0 0 0 0 √-2 -√-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ14 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 2 2 -2 -2 -2 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ15 4 4 4 4 2 2 1 -2 0 0 0 0 1 1 -2 1 -2 -2 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 4 4 0 0 -2 1 2 2 0 0 -2 -2 1 -2 1 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ17 4 4 4 4 0 0 -2 1 -2 -2 0 0 -2 -2 1 -2 1 1 0 0 0 0 0 0 0 0 1 1 1 1 orthogonal lifted from S3≀C2 ρ18 4 4 4 4 -2 -2 1 -2 0 0 0 0 1 1 -2 1 -2 -2 1 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ19 4 4 -4 -4 -2 2 1 -2 0 0 0 0 -1 -1 2 1 -2 2 -1 -1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ20 4 4 -4 -4 0 0 -2 1 -2 2 0 0 2 2 -1 -2 1 -1 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×S3≀C2 ρ21 4 4 -4 -4 0 0 -2 1 2 -2 0 0 2 2 -1 -2 1 -1 0 0 0 0 0 0 0 0 1 1 -1 -1 orthogonal lifted from C2×S3≀C2 ρ22 4 4 -4 -4 2 -2 1 -2 0 0 0 0 -1 -1 2 1 -2 2 1 1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ23 4 -4 4 -4 0 0 -2 1 0 0 0 0 2 -2 1 2 -1 -1 0 0 0 0 0 0 0 0 √3 -√3 -√3 √3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ24 4 -4 -4 4 0 0 -2 1 0 0 0 0 -2 2 -1 2 -1 1 0 0 0 0 0 0 0 0 -√3 √3 -√3 √3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ25 4 -4 4 -4 0 0 -2 1 0 0 0 0 2 -2 1 2 -1 -1 0 0 0 0 0 0 0 0 -√3 √3 √3 -√3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ26 4 -4 -4 4 0 0 -2 1 0 0 0 0 -2 2 -1 2 -1 1 0 0 0 0 0 0 0 0 √3 -√3 √3 -√3 symplectic lifted from C32⋊2SD16, Schur index 2 ρ27 4 -4 4 -4 0 0 1 -2 0 0 0 0 -1 1 -2 -1 2 2 -√-3 √-3 √-3 -√-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ28 4 -4 -4 4 0 0 1 -2 0 0 0 0 1 -1 2 -1 2 -2 -√-3 √-3 -√-3 √-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ29 4 -4 4 -4 0 0 1 -2 0 0 0 0 -1 1 -2 -1 2 2 √-3 -√-3 -√-3 √-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16 ρ30 4 -4 -4 4 0 0 1 -2 0 0 0 0 1 -1 2 -1 2 -2 √-3 -√-3 √-3 -√-3 0 0 0 0 0 0 0 0 complex lifted from C32⋊2SD16

Smallest permutation representation of C2×C322SD16
On 48 points
Generators in S48
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 45 31)(3 25 47)(5 41 27)(7 29 43)(9 18 35)(11 37 20)(13 22 39)(15 33 24)
(2 46 32)(4 26 48)(6 42 28)(8 30 44)(10 19 36)(12 38 21)(14 23 40)(16 34 17)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 41)(10 44)(11 47)(12 42)(13 45)(14 48)(15 43)(16 46)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)

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

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

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

Matrix representation of C2×C322SD16 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 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 5 40 1 0 0 0 5 40 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 23 38 72 72 0 0 0 0 1 0
,
 60 50 0 0 0 0 55 13 0 0 0 0 0 0 23 38 71 72 0 0 0 0 72 1 0 0 49 5 45 5 0 0 6 18 45 5
,
 72 0 0 0 0 0 17 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 4 60 30 60 0 0 70 4 13 43

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,5,5,0,0,72,72,40,40,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,0,0,0,0,1,38,0,0,0,0,0,72,1,0,0,0,0,72,0],[60,55,0,0,0,0,50,13,0,0,0,0,0,0,23,0,49,6,0,0,38,0,5,18,0,0,71,72,45,45,0,0,72,1,5,5],[72,17,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,70,0,0,1,0,60,4,0,0,0,0,30,13,0,0,0,0,60,43] >;

C2×C322SD16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2{\rm SD}_{16}
% in TeX

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

G:=SmallGroup(288,886);
// by ID

G=gap.SmallGroup(288,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,675,346,80,2693,2028,362,797,1203]);
// 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=c^-1,e*b*e=b^-1,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations

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