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## G = C2×C32⋊Q16order 288 = 25·32

### Direct product of C2 and C32⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C2×C32⋊Q16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2Q8 — C32⋊Q16 — C2×C32⋊Q16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C2×C32⋊Q16
 Upper central C1 — C22

Generators and relations for C2×C32⋊Q16
G = < a,b,c,d,e | a2=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=c, dcd-1=b-1, ece-1=b, ede-1=d-1 >

Subgroups: 432 in 98 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×6], C22, C6 [×6], C8 [×2], C2×C4 [×3], Q8 [×6], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], C2×C8, Q16 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], Dic6 [×8], C2×Dic3 [×4], C2×C12 [×2], C2×Q16, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C2×Dic6 [×2], C322C8 [×2], C322Q8 [×4], C322Q8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C32⋊Q16 [×4], C2×C322C8, C2×C322Q8 [×2], C2×C32⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, C2×Q16, S3≀C2, C32⋊Q16 [×2], C2×S3≀C2, C2×C32⋊Q16

Character table of C2×C32⋊Q16

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 4 4 12 12 12 12 18 18 4 4 4 4 4 4 18 18 18 18 12 12 12 12 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 2 2 0 0 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 2 2 0 0 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 2 2 0 0 0 0 0 0 -2 -2 -2 -2 2 2 √2 √2 -√2 -√2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 2 -2 -2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 2 2 -√2 -√2 √2 √2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 2 -2 -2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 4 -4 -4 4 -2 1 -2 0 2 0 0 0 -2 -1 2 1 -1 2 0 0 0 0 -1 0 1 1 0 0 0 -1 orthogonal lifted from C2×S3≀C2 ρ16 4 -4 -4 4 1 -2 0 2 0 -2 0 0 1 2 -1 -2 2 -1 0 0 0 0 0 1 0 0 -1 -1 1 0 orthogonal lifted from C2×S3≀C2 ρ17 4 -4 -4 4 -2 1 2 0 -2 0 0 0 -2 -1 2 1 -1 2 0 0 0 0 1 0 -1 -1 0 0 0 1 orthogonal lifted from C2×S3≀C2 ρ18 4 4 4 4 1 -2 0 2 0 2 0 0 1 -2 1 -2 -2 1 0 0 0 0 0 -1 0 0 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ19 4 4 4 4 1 -2 0 -2 0 -2 0 0 1 -2 1 -2 -2 1 0 0 0 0 0 1 0 0 1 1 1 0 orthogonal lifted from S3≀C2 ρ20 4 4 4 4 -2 1 2 0 2 0 0 0 -2 1 -2 1 1 -2 0 0 0 0 -1 0 -1 -1 0 0 0 -1 orthogonal lifted from S3≀C2 ρ21 4 -4 -4 4 1 -2 0 -2 0 2 0 0 1 2 -1 -2 2 -1 0 0 0 0 0 -1 0 0 1 1 -1 0 orthogonal lifted from C2×S3≀C2 ρ22 4 4 4 4 -2 1 -2 0 -2 0 0 0 -2 1 -2 1 1 -2 0 0 0 0 1 0 1 1 0 0 0 1 orthogonal lifted from S3≀C2 ρ23 4 4 -4 -4 -2 1 0 0 0 0 0 0 2 1 -2 -1 -1 2 0 0 0 0 -√3 0 -√3 √3 0 0 0 √3 symplectic lifted from C32⋊Q16, Schur index 2 ρ24 4 -4 4 -4 1 -2 0 0 0 0 0 0 -1 2 -1 2 -2 1 0 0 0 0 0 √3 0 0 √3 -√3 -√3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ25 4 -4 4 -4 -2 1 0 0 0 0 0 0 2 -1 2 -1 1 -2 0 0 0 0 √3 0 -√3 √3 0 0 0 -√3 symplectic lifted from C32⋊Q16, Schur index 2 ρ26 4 4 -4 -4 1 -2 0 0 0 0 0 0 -1 -2 1 2 2 -1 0 0 0 0 0 √3 0 0 -√3 √3 -√3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ27 4 -4 4 -4 1 -2 0 0 0 0 0 0 -1 2 -1 2 -2 1 0 0 0 0 0 -√3 0 0 -√3 √3 √3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ28 4 4 -4 -4 1 -2 0 0 0 0 0 0 -1 -2 1 2 2 -1 0 0 0 0 0 -√3 0 0 √3 -√3 √3 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ29 4 -4 4 -4 -2 1 0 0 0 0 0 0 2 -1 2 -1 1 -2 0 0 0 0 -√3 0 √3 -√3 0 0 0 √3 symplectic lifted from C32⋊Q16, Schur index 2 ρ30 4 4 -4 -4 -2 1 0 0 0 0 0 0 2 1 -2 -1 -1 2 0 0 0 0 √3 0 √3 -√3 0 0 0 -√3 symplectic lifted from C32⋊Q16, Schur index 2

Smallest permutation representation of C2×C32⋊Q16
On 96 points
Generators in S96
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 64)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 92)(34 93)(35 94)(36 95)(37 96)(38 89)(39 90)(40 91)(41 69)(42 70)(43 71)(44 72)(45 65)(46 66)(47 67)(48 68)
(2 32 10)(4 12 26)(6 28 14)(8 16 30)(17 91 47)(19 41 93)(21 95 43)(23 45 89)(34 58 69)(36 71 60)(38 62 65)(40 67 64)(50 88 75)(52 77 82)(54 84 79)(56 73 86)
(1 31 9)(3 11 25)(5 27 13)(7 15 29)(18 48 92)(20 94 42)(22 44 96)(24 90 46)(33 57 68)(35 70 59)(37 61 72)(39 66 63)(49 87 74)(51 76 81)(53 83 78)(55 80 85)
(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 64 5 60)(2 63 6 59)(3 62 7 58)(4 61 8 57)(9 67 13 71)(10 66 14 70)(11 65 15 69)(12 72 16 68)(17 87 21 83)(18 86 22 82)(19 85 23 81)(20 84 24 88)(25 38 29 34)(26 37 30 33)(27 36 31 40)(28 35 32 39)(41 55 45 51)(42 54 46 50)(43 53 47 49)(44 52 48 56)(73 96 77 92)(74 95 78 91)(75 94 79 90)(76 93 80 89)

G:=sub<Sym(96)| (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,64)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68), (2,32,10)(4,12,26)(6,28,14)(8,16,30)(17,91,47)(19,41,93)(21,95,43)(23,45,89)(34,58,69)(36,71,60)(38,62,65)(40,67,64)(50,88,75)(52,77,82)(54,84,79)(56,73,86), (1,31,9)(3,11,25)(5,27,13)(7,15,29)(18,48,92)(20,94,42)(22,44,96)(24,90,46)(33,57,68)(35,70,59)(37,61,72)(39,66,63)(49,87,74)(51,76,81)(53,83,78)(55,80,85), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,64,5,60)(2,63,6,59)(3,62,7,58)(4,61,8,57)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,87,21,83)(18,86,22,82)(19,85,23,81)(20,84,24,88)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89)>;

G:=Group( (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,64)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68), (2,32,10)(4,12,26)(6,28,14)(8,16,30)(17,91,47)(19,41,93)(21,95,43)(23,45,89)(34,58,69)(36,71,60)(38,62,65)(40,67,64)(50,88,75)(52,77,82)(54,84,79)(56,73,86), (1,31,9)(3,11,25)(5,27,13)(7,15,29)(18,48,92)(20,94,42)(22,44,96)(24,90,46)(33,57,68)(35,70,59)(37,61,72)(39,66,63)(49,87,74)(51,76,81)(53,83,78)(55,80,85), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,64,5,60)(2,63,6,59)(3,62,7,58)(4,61,8,57)(9,67,13,71)(10,66,14,70)(11,65,15,69)(12,72,16,68)(17,87,21,83)(18,86,22,82)(19,85,23,81)(20,84,24,88)(25,38,29,34)(26,37,30,33)(27,36,31,40)(28,35,32,39)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(73,96,77,92)(74,95,78,91)(75,94,79,90)(76,93,80,89) );

G=PermutationGroup([(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,81),(8,82),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,64),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,80),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,92),(34,93),(35,94),(36,95),(37,96),(38,89),(39,90),(40,91),(41,69),(42,70),(43,71),(44,72),(45,65),(46,66),(47,67),(48,68)], [(2,32,10),(4,12,26),(6,28,14),(8,16,30),(17,91,47),(19,41,93),(21,95,43),(23,45,89),(34,58,69),(36,71,60),(38,62,65),(40,67,64),(50,88,75),(52,77,82),(54,84,79),(56,73,86)], [(1,31,9),(3,11,25),(5,27,13),(7,15,29),(18,48,92),(20,94,42),(22,44,96),(24,90,46),(33,57,68),(35,70,59),(37,61,72),(39,66,63),(49,87,74),(51,76,81),(53,83,78),(55,80,85)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,64,5,60),(2,63,6,59),(3,62,7,58),(4,61,8,57),(9,67,13,71),(10,66,14,70),(11,65,15,69),(12,72,16,68),(17,87,21,83),(18,86,22,82),(19,85,23,81),(20,84,24,88),(25,38,29,34),(26,37,30,33),(27,36,31,40),(28,35,32,39),(41,55,45,51),(42,54,46,50),(43,53,47,49),(44,52,48,56),(73,96,77,92),(74,95,78,91),(75,94,79,90),(76,93,80,89)])

Matrix representation of C2×C32⋊Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 55 48 0 0 0 0 25 59 0 0 0 0 0 0 0 0 14 19 0 0 0 0 5 59 0 0 66 59 0 0 0 0 14 7 0 0
,
 57 4 0 0 0 0 27 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[55,25,0,0,0,0,48,59,0,0,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,14,5,0,0,0,0,19,59,0,0],[57,27,0,0,0,0,4,16,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C32⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes Q_{16}
% in TeX

G:=Group("C2xC3^2:Q16");
// GroupNames label

G:=SmallGroup(288,888);
// by ID

G=gap.SmallGroup(288,888);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=1,e^2=d^4,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^-1=c,d*c*d^-1=b^-1,e*c*e^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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