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G = C6.SD16order 96 = 25·3

2nd non-split extension by C6 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.2D4, C6.4Q16, Dic63C4, C4.10D12, C6.5SD16, C4⋊C4.3S3, C4.2(C4×S3), C12.4(C2×C4), (C2×C4).36D6, (C2×C6).31D4, C2.6(D6⋊C4), C31(Q8⋊C4), C6.4(C22⋊C4), C2.2(D4.S3), (C2×Dic6).5C2, C2.2(C3⋊Q16), (C2×C12).11C22, C22.15(C3⋊D4), (C2×C3⋊C8).3C2, (C3×C4⋊C4).3C2, SmallGroup(96,17)

Series: Derived Chief Lower central Upper central

C1C12 — C6.SD16
C1C3C6C2×C6C2×C12C2×Dic6 — C6.SD16
C3C6C12 — C6.SD16
C1C22C2×C4C4⋊C4

Generators and relations for C6.SD16
 G = < a,b,c | a6=b8=1, c2=a3b4, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

4C4
6C4
6C4
2C2×C4
3Q8
3Q8
6C8
6Q8
6C2×C4
2Dic3
2Dic3
4C12
3C2×C8
3C2×Q8
2C3⋊C8
2C2×C12
2C2×Dic3
2Dic6
3Q8⋊C4

Character table of C6.SD16

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F
 size 11112224412122226666444444
ρ1111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-11-1-11-1-1    linear of order 2
ρ3111111111-1-1111-1-1-1-1111111    linear of order 2
ρ41111111-1-1-1-111111111-1-11-1-1    linear of order 2
ρ511-1-11-11i-i1-1-1-11-ii-ii-1ii1-i-i    linear of order 4
ρ611-1-11-11i-i-11-1-11i-ii-i-1ii1-i-i    linear of order 4
ρ711-1-11-11-ii-11-1-11-ii-ii-1-i-i1ii    linear of order 4
ρ811-1-11-11-ii1-1-1-11i-ii-i-1-i-i1ii    linear of order 4
ρ92222-1222200-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022222-2-200002220000-200-200    orthogonal lifted from D4
ρ112222-122-2-200-1-1-10000-111-111    orthogonal lifted from D6
ρ1222-2-222-20000-2-220000200-200    orthogonal lifted from D4
ρ1322-2-2-12-2000011-10000-13-31-33    orthogonal lifted from D12
ρ1422-2-2-12-2000011-10000-1-3313-3    orthogonal lifted from D12
ρ152-2-222000000-22-22-2-22000000    symplectic lifted from Q16, Schur index 2
ρ162-2-222000000-22-2-222-2000000    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-1-22-2i2i0011-100001ii-1-i-i    complex lifted from C4×S3
ρ182222-1-2-20000-1-1-100001-3--31-3--3    complex lifted from C3⋊D4
ρ192-22-220000002-2-2--2--2-2-2000000    complex lifted from SD16
ρ2022-2-2-1-222i-2i0011-100001-i-i-1ii    complex lifted from C4×S3
ρ212222-1-2-20000-1-1-100001--3-31--3-3    complex lifted from C3⋊D4
ρ222-22-220000002-2-2-2-2--2--2000000    complex lifted from SD16
ρ234-4-44-20000002-220000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ244-44-4-2000000-2220000000000    symplectic lifted from D4.S3, Schur index 2

Smallest permutation representation of C6.SD16
Regular action on 96 points
Generators in S96
(1 75 93 23 27 61)(2 62 28 24 94 76)(3 77 95 17 29 63)(4 64 30 18 96 78)(5 79 89 19 31 57)(6 58 32 20 90 80)(7 73 91 21 25 59)(8 60 26 22 92 74)(9 48 34 55 67 83)(10 84 68 56 35 41)(11 42 36 49 69 85)(12 86 70 50 37 43)(13 44 38 51 71 87)(14 88 72 52 39 45)(15 46 40 53 65 81)(16 82 66 54 33 47)
(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 55 19 13)(2 16 20 50)(3 53 21 11)(4 14 22 56)(5 51 23 9)(6 12 24 54)(7 49 17 15)(8 10 18 52)(25 42 77 65)(26 68 78 45)(27 48 79 71)(28 66 80 43)(29 46 73 69)(30 72 74 41)(31 44 75 67)(32 70 76 47)(33 58 86 94)(34 89 87 61)(35 64 88 92)(36 95 81 59)(37 62 82 90)(38 93 83 57)(39 60 84 96)(40 91 85 63)

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

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

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

C6.SD16 is a maximal subgroup of
C12⋊Q8⋊C2  Dic6.D4  (C2×C8).200D6  D4⋊(C4×S3)  D42S3⋊C4  D43D12  D4.D12  D12.D4  Dic3.1Q16  Dic3⋊Q16  (C2×Q8).36D6  S3×Q8⋊C4  (S3×Q8)⋊C4  Q83D12  D6⋊Q16  Dic3⋊SD16  Dic38SD16  Dic129C4  Dic6⋊Q8  Dic6.Q8  D6.2SD16  C88D12  C8.2D12  C6.(C4○D8)  Dic35Q16  Dic3.Q16  Dic6.2Q8  D6.2Q16  C83D12  D62Q16  C2.D87S3  C24⋊C2⋊C4  C4○D12⋊C4  C4⋊C4.230D6  C4⋊C4.231D6  C4⋊C4.233D6  C4.(C2×D12)  C4⋊C4.237D6  D4.1D12  C4×D4.S3  C42.51D6  D4.2D12  Q8.6D12  C4×C3⋊Q16  C42.59D6  C127Q16  D1217D4  Dic617D4  C3⋊C823D4  C3⋊C85D4  D12.37D4  Dic6.37D4  C3⋊C8.29D4  C3⋊C8.6D4  Dic6.4Q8  C42.216D6  C42.71D6  C42.82D6  Dic65Q8  C12.Q16  Dic66Q8  C18.Q16  Dic6⋊Dic3  C12.73D12  C62.114D4  Dic6⋊Dic5  Dic3012C4  Dic309C4  Dic6⋊F5
C6.SD16 is a maximal quotient of
C4⋊Dic3⋊C4  C4.Dic12  Dic62C8  C12.2D8  C12.C42  C18.Q16  Dic6⋊Dic3  C12.73D12  C62.114D4  Dic6⋊Dic5  Dic3012C4  Dic309C4  Dic6⋊F5

Matrix representation of C6.SD16 in GL5(𝔽73)

720000
0727200
01000
000720
000072
,
270000
0546800
0141900
00004
0005512
,
270000
0306000
0134300
0001166
0003862

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,54,14,0,0,0,68,19,0,0,0,0,0,0,55,0,0,0,4,12],[27,0,0,0,0,0,30,13,0,0,0,60,43,0,0,0,0,0,11,38,0,0,0,66,62] >;

C6.SD16 in GAP, Magma, Sage, TeX

C_6.{\rm SD}_{16}
% in TeX

G:=Group("C6.SD16");
// GroupNames label

G:=SmallGroup(96,17);
// by ID

G=gap.SmallGroup(96,17);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=1,c^2=a^3*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

Export

Subgroup lattice of C6.SD16 in TeX
Character table of C6.SD16 in TeX

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