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G = C2×C32⋊Q16order 288 = 25·32

Direct product of C2 and C32⋊Q16

direct product, non-abelian, soluble, monomial

Aliases: C2×C32⋊Q16, C62.16D4, (C3×C6)⋊Q16, C322(C2×Q16), C22.16S3≀C2, C3⋊Dic3.34D4, C3⋊Dic3.12C23, C322C8.8C22, C322Q8.8C22, C2.21(C2×S3≀C2), (C3×C6).21(C2×D4), (C2×C322C8).6C2, (C2×C322Q8).5C2, (C2×C3⋊Dic3).99C22, SmallGroup(288,888)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C2×C32⋊Q16
C1C32C3×C6C3⋊Dic3C322Q8C32⋊Q16 — C2×C32⋊Q16
C32C3×C6C3⋊Dic3 — C2×C32⋊Q16
C1C22

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 12A2B2C3A3B4A4B4C4D4E4F6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H
 size 111144121212121818444444181818181212121212121212
ρ1111111111111111111111111111111    trivial
ρ21-1-1111-1-1111-11-1-11-1-1-11-1111-1-1-1-111    linear of order 2
ρ31-1-1111-111-11-11-1-11-1-11-11-11-1-1-111-11    linear of order 2
ρ41111111-11-111111111-1-1-1-11-111-1-1-11    linear of order 2
ρ51-1-111111-1-11-11-1-11-1-1-11-11-1-11111-1-1    linear of order 2
ρ6111111-1-1-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-11-1111111111-1-1-1-1-11-1-1111-1    linear of order 2
ρ81-1-11111-1-111-11-1-11-1-11-11-1-1111-1-11-1    linear of order 2
ρ92222220000-2-2222222000000000000    orthogonal lifted from D4
ρ102-2-22220000-222-2-22-2-2000000000000    orthogonal lifted from D4
ρ112-22-222000000-2-2-2-22222-2-200000000    symplectic lifted from Q16, Schur index 2
ρ1222-2-222000000-222-2-2-2-222-200000000    symplectic lifted from Q16, Schur index 2
ρ132-22-222000000-2-2-2-222-2-22200000000    symplectic lifted from Q16, Schur index 2
ρ1422-2-222000000-222-2-2-22-2-2200000000    symplectic lifted from Q16, Schur index 2
ρ154-4-44-21-202000-2-121-120000-1011000-1    orthogonal lifted from C2×S3≀C2
ρ164-4-441-2020-20012-1-22-100000100-1-110    orthogonal lifted from C2×S3≀C2
ρ174-4-44-2120-2000-2-121-12000010-1-10001    orthogonal lifted from C2×S3≀C2
ρ1844441-20202001-21-2-2100000-100-1-1-10    orthogonal lifted from S3≀C2
ρ1944441-20-20-2001-21-2-21000001001110    orthogonal lifted from S3≀C2
ρ204444-21202000-21-211-20000-10-1-1000-1    orthogonal lifted from S3≀C2
ρ214-4-441-20-2020012-1-22-100000-10011-10    orthogonal lifted from C2×S3≀C2
ρ224444-21-20-2000-21-211-2000010110001    orthogonal lifted from S3≀C2
ρ2344-4-4-2100000021-2-1-120000-30-330003    symplectic lifted from C32⋊Q16, Schur index 2
ρ244-44-41-2000000-12-12-21000003003-3-30    symplectic lifted from C32⋊Q16, Schur index 2
ρ254-44-4-210000002-12-11-2000030-33000-3    symplectic lifted from C32⋊Q16, Schur index 2
ρ2644-4-41-2000000-1-2122-100000300-33-30    symplectic lifted from C32⋊Q16, Schur index 2
ρ274-44-41-2000000-12-12-2100000-300-3330    symplectic lifted from C32⋊Q16, Schur index 2
ρ2844-4-41-2000000-1-2122-100000-3003-330    symplectic lifted from C32⋊Q16, Schur index 2
ρ294-44-4-210000002-12-11-20000-303-30003    symplectic lifted from C32⋊Q16, Schur index 2
ρ3044-4-4-2100000021-2-1-120000303-3000-3    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)

100000
010000
0072000
0007200
0000720
0000072
,
100000
010000
001000
000100
000001
00007272
,
100000
010000
000100
00727200
000010
000001
,
55480000
25590000
00001419
0000559
00665900
0014700
,
5740000
27160000
000010
000001
0072000
0007200

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

Export

Character table of C2×C32⋊Q16 in TeX

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