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G = C42.82D6order 192 = 26·3

82nd non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.82D6, C4⋊Q87S3, C4⋊C4.85D6, (C2×C12).296D4, C12.84(C4○D4), C6.SD1642C2, C6.99(C8⋊C22), C427S3.9C2, C6.D8.16C2, (C2×C12).407C23, (C4×C12).136C22, C42.S315C2, C4.17(Q83S3), C6.59(C4.4D4), C6.98(C8.C22), C2.20(D126C22), (C2×D12).110C22, C2.19(Q8.11D6), C2.12(C12.23D4), C35(C42.28C22), (C2×Dic6).114C22, (C3×C4⋊Q8)⋊7C2, (C2×C6).538(C2×D4), (C2×C4).74(C3⋊D4), (C2×C3⋊C8).139C22, (C3×C4⋊C4).132C22, (C2×C4).504(C22×S3), C22.210(C2×C3⋊D4), SmallGroup(192,648)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.82D6
C1C3C6C12C2×C12C2×D12C427S3 — C42.82D6
C3C6C2×C12 — C42.82D6
C1C22C42C4⋊Q8

Generators and relations for C42.82D6
 G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 304 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], S3, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4, C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C22×S3, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C2×C3⋊C8 [×2], D6⋊C4 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, C42.28C22, C42.S3, C6.D8 [×2], C6.SD16 [×2], C427S3, C3×C4⋊Q8, C42.82D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C8⋊C22, C8.C22, Q83S3 [×2], C2×C3⋊D4, C42.28C22, D126C22, Q8.11D6, C12.23D4, C42.82D6

Character table of C42.82D6

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H12I12J
 size 111124222448824222121212124444448888
ρ1111111111111111111111111111111    trivial
ρ21111-111111-1-1-11111111111111-1-1-1-1    linear of order 2
ρ31111-1111-1-1-111111-11-11-111-1-1-11-1-11    linear of order 2
ρ411111111-1-11-1-1111-11-11-111-1-1-1-111-1    linear of order 2
ρ51111111111-1-11111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ61111-11111111-1111-1-1-1-11111111111    linear of order 2
ρ71111-1111-1-11-111111-11-1-111-1-1-1-111-1    linear of order 2
ρ811111111-1-1-11-11111-11-1-111-1-1-11-1-11    linear of order 2
ρ9222202-2-22-20002220000-2-2-222-20000    orthogonal lifted from D4
ρ1022220-122-2-22-20-1-1-100001-1-11111-1-11    orthogonal lifted from D6
ρ11222202-2-2-2200022200002-2-2-2-220000    orthogonal lifted from D4
ρ1222220-12222-2-20-1-1-10000-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1322220-12222220-1-1-10000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422220-122-2-2-220-1-1-100001-1-1111-111-1    orthogonal lifted from D6
ρ1522220-1-2-22-2000-1-1-10000111-1-11--3--3-3-3    complex lifted from C3⋊D4
ρ1622220-1-2-2-22000-1-1-10000-11111-1--3-3--3-3    complex lifted from C3⋊D4
ρ1722220-1-2-2-22000-1-1-10000-11111-1-3--3-3--3    complex lifted from C3⋊D4
ρ1822220-1-2-22-2000-1-1-10000111-1-11-3-3--3--3    complex lifted from C3⋊D4
ρ192-2-22022-200000-2-222i0-2i00-220000000    complex lifted from C4○D4
ρ202-2-2202-2200000-2-2202i0-2i02-20000000    complex lifted from C4○D4
ρ212-2-2202-2200000-2-220-2i02i02-20000000    complex lifted from C4○D4
ρ222-2-22022-200000-2-22-2i02i00-220000000    complex lifted from C4○D4
ρ234-4-440-24-40000022-2000002-20000000    orthogonal lifted from Q83S3, Schur index 2
ρ244-44-40400000004-4-400000000000000    orthogonal lifted from C8⋊C22
ρ254-4-440-2-440000022-200000-220000000    orthogonal lifted from Q83S3, Schur index 2
ρ2644-4-4040000000-44-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-44-40-20000000-22200002-30000-2-30000    complex lifted from D126C22
ρ2844-4-40-200000002-2200000002-3-2-300000    complex lifted from Q8.11D6
ρ2944-4-40-200000002-220000000-2-32-300000    complex lifted from Q8.11D6
ρ304-44-40-20000000-2220000-2-300002-30000    complex lifted from D126C22

Smallest permutation representation of C42.82D6
On 96 points
Generators in S96
(1 55 29 80)(2 81 30 56)(3 57 31 82)(4 83 32 58)(5 59 33 84)(6 73 34 60)(7 49 35 74)(8 75 36 50)(9 51 25 76)(10 77 26 52)(11 53 27 78)(12 79 28 54)(13 61 46 87)(14 88 47 62)(15 63 48 89)(16 90 37 64)(17 65 38 91)(18 92 39 66)(19 67 40 93)(20 94 41 68)(21 69 42 95)(22 96 43 70)(23 71 44 85)(24 86 45 72)
(1 46 35 19)(2 20 36 47)(3 48 25 21)(4 22 26 37)(5 38 27 23)(6 24 28 39)(7 40 29 13)(8 14 30 41)(9 42 31 15)(10 16 32 43)(11 44 33 17)(12 18 34 45)(49 93 80 61)(50 62 81 94)(51 95 82 63)(52 64 83 96)(53 85 84 65)(54 66 73 86)(55 87 74 67)(56 68 75 88)(57 89 76 69)(58 70 77 90)(59 91 78 71)(60 72 79 92)
(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 6 13 45 35 28 40 18)(2 17 41 27 36 44 14 5)(3 4 15 43 25 26 42 16)(7 12 19 39 29 34 46 24)(8 23 47 33 30 38 20 11)(9 10 21 37 31 32 48 22)(49 60 67 86 80 79 87 66)(50 65 88 78 81 85 68 59)(51 58 69 96 82 77 89 64)(52 63 90 76 83 95 70 57)(53 56 71 94 84 75 91 62)(54 61 92 74 73 93 72 55)

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

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

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

Matrix representation of C42.82D6 in GL6(𝔽73)

5460000
37190000
00003060
00001343
00431300
00603000
,
100000
010000
000010
000001
0072000
0007200
,
2700000
25460000
0042313142
0042113162
0031423142
0031623162
,
2700000
0270000
0031423142
0011421142
0042313142
0062311142

G:=sub<GL(6,GF(73))| [54,37,0,0,0,0,6,19,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,30,13,0,0,0,0,60,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[27,25,0,0,0,0,0,46,0,0,0,0,0,0,42,42,31,31,0,0,31,11,42,62,0,0,31,31,31,31,0,0,42,62,42,62],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,31,11,42,62,0,0,42,42,31,31,0,0,31,11,31,11,0,0,42,42,42,42] >;

C42.82D6 in GAP, Magma, Sage, TeX

C_4^2._{82}D_6
% in TeX

G:=Group("C4^2.82D6");
// GroupNames label

G:=SmallGroup(192,648);
// by ID

G=gap.SmallGroup(192,648);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

Export

Character table of C42.82D6 in TeX

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