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G = C6.Q16order 96 = 25·3

1st non-split extension by C6 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.6D8, C6.3Q16, C12.1Q8, C4.1Dic6, C3⋊C81C4, C4⋊C4.1S3, C31(C2.D8), C6.2(C4⋊C4), C4.11(C4×S3), C12.1(C2×C4), (C2×C6).28D4, (C2×C4).33D6, C2.1(D4⋊S3), C4⋊Dic3.8C2, (C2×C12).8C22, C2.1(C3⋊Q16), C2.3(Dic3⋊C4), C22.12(C3⋊D4), (C2×C3⋊C8).1C2, (C3×C4⋊C4).1C2, SmallGroup(96,14)

Series: Derived Chief Lower central Upper central

C1C12 — C6.Q16
C1C3C6C2×C6C2×C12C2×C3⋊C8 — C6.Q16
C3C6C12 — C6.Q16
C1C22C2×C4C4⋊C4

Generators and relations for C6.Q16
 G = < a,b,c | a12=b4=1, c2=a9b2, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >

4C4
12C4
2C2×C4
3C8
3C8
6C2×C4
4C12
4Dic3
3C2×C8
3C4⋊C4
2C2×Dic3
2C2×C12
3C2.D8

Character table of C6.Q16

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F
 size 11112224412122226666444444
ρ1111111111111111111111111    trivial
ρ21111111-1-1-1-11111111-1-1-111-1    linear of order 2
ρ3111111111-1-1111-1-1-1-1111111    linear of order 2
ρ41111111-1-111111-1-1-1-1-1-1-111-1    linear of order 2
ρ511-1-11-11i-i-ii-11-1-111-1i-i-i1-1i    linear of order 4
ρ611-1-11-11-iii-i-11-1-111-1-iii1-1-i    linear of order 4
ρ711-1-11-11i-ii-i-11-11-1-11i-i-i1-1i    linear of order 4
ρ811-1-11-11-ii-ii-11-11-1-11-iii1-1-i    linear of order 4
ρ92222-1222200-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-122-2-200-1-1-10000111-1-11    orthogonal lifted from D6
ρ1122222-2-200002220000000-2-20    orthogonal lifted from D4
ρ122-22-22000000-2-222-22-2000000    orthogonal lifted from D8
ρ132-22-22000000-2-22-22-22000000    orthogonal lifted from D8
ρ1422-2-222-20000-22-20000000-220    symplectic lifted from Q8, Schur index 2
ρ152-2-2220000002-2-2-2-222000000    symplectic lifted from Q16, Schur index 2
ρ162-2-2220000002-2-222-2-2000000    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-12-200001-110000-3-331-13    symplectic lifted from Dic6, Schur index 2
ρ1822-2-2-12-200001-11000033-31-1-3    symplectic lifted from Dic6, Schur index 2
ρ1922-2-2-1-222i-2i001-110000-iii-11-i    complex lifted from C4×S3
ρ2022-2-2-1-22-2i2i001-110000i-i-i-11i    complex lifted from C4×S3
ρ212222-1-2-20000-1-1-10000-3--3-311--3    complex lifted from C3⋊D4
ρ222222-1-2-20000-1-1-10000--3-3--311-3    complex lifted from C3⋊D4
ρ234-44-4-200000022-20000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ244-4-44-2000000-2220000000000    symplectic lifted from C3⋊Q16, Schur index 2

Smallest permutation representation of C6.Q16
Regular action on 96 points
Generators in S96
(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 38 64 22)(2 45 65 17)(3 40 66 24)(4 47 67 19)(5 42 68 14)(6 37 69 21)(7 44 70 16)(8 39 71 23)(9 46 72 18)(10 41 61 13)(11 48 62 20)(12 43 63 15)(25 84 89 58)(26 79 90 53)(27 74 91 60)(28 81 92 55)(29 76 93 50)(30 83 94 57)(31 78 95 52)(32 73 96 59)(33 80 85 54)(34 75 86 49)(35 82 87 56)(36 77 88 51)
(1 80 61 51 7 74 67 57)(2 73 62 56 8 79 68 50)(3 78 63 49 9 84 69 55)(4 83 64 54 10 77 70 60)(5 76 65 59 11 82 71 53)(6 81 66 52 12 75 72 58)(13 85 38 30 19 91 44 36)(14 90 39 35 20 96 45 29)(15 95 40 28 21 89 46 34)(16 88 41 33 22 94 47 27)(17 93 42 26 23 87 48 32)(18 86 43 31 24 92 37 25)

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

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

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

C6.Q16 is a maximal subgroup of
Dic34D8  D4.S3⋊C4  Dic3.D8  D4.Dic6  D6.D8  D6⋊D8  D6⋊C811C2  C3⋊C81D4  Dic34Q16  Q83Dic6  Q8.4Dic6  D6.Q16  C3⋊(C8⋊D4)  D61Q16  C8⋊Dic3⋊C2  Q83(C4×S3)  C243Q8  Dic6.Q8  C8.8Dic6  (S3×C8)⋊C4  C8⋊(C4×S3)  C4.Q8⋊S3  C6.(C4○D8)  D12.Q8  C242Q8  Dic3.Q16  C244Q8  S3×C2.D8  C8⋊S3⋊C4  D6.5D8  D6.2Q16  D122Q8  C4⋊C4.225D6  (C2×C6).40D8  C4⋊C4.230D6  C4⋊C4.232D6  C4⋊C4.233D6  C4⋊C4.234D6  C4⋊C4.236D6  C12.50D8  D4.3Dic6  C4×D4⋊S3  C42.51D6  Q85Dic6  Q8.5Dic6  C42.56D6  C4×C3⋊Q16  (C2×C6).D8  C6.Q16⋊C2  C3⋊C822D4  C3⋊C85D4  (C2×C6).Q16  (C2×Q8).51D6  C3⋊C86D4  C3⋊C8.29D4  Dic6.4Q8  C42.68D6  C42.215D6  D12.4Q8  C12.17D8  C42.76D6  D126Q8  Dic65Q8  C36.Q8  C6.18D24  C12.8Dic6  C12.9Dic6  C30.20D8  C60.5Q8  C60.1Q8  Dic5.4Dic6
C6.Q16 is a maximal quotient of
C12.53D8  C6.6D16  C6.SD32  C24.7Q8  C12.C42  C36.Q8  C6.18D24  C12.8Dic6  C12.9Dic6  C30.20D8  C60.5Q8  C60.1Q8  Dic5.4Dic6

Matrix representation of C6.Q16 in GL4(𝔽73) generated by

0100
727200
00722
00721
,
71400
596600
003641
002037
,
603000
431300
003241
00160
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,72,72,0,0,2,1],[7,59,0,0,14,66,0,0,0,0,36,20,0,0,41,37],[60,43,0,0,30,13,0,0,0,0,32,16,0,0,41,0] >;

C6.Q16 in GAP, Magma, Sage, TeX

C_6.Q_{16}
% in TeX

G:=Group("C6.Q16");
// GroupNames label

G:=SmallGroup(96,14);
// by ID

G=gap.SmallGroup(96,14);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=a^9*b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations

Export

Subgroup lattice of C6.Q16 in TeX
Character table of C6.Q16 in TeX

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