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G = C42.80D6order 192 = 26·3

80th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.80D6, C4⋊Q84S3, C3⋊C8.8D4, C4.18(S3×D4), C12.38(C2×D4), C33(C8.2D4), (C2×Q8).70D6, (C2×C12).295D4, C6.24(C41D4), C427S3.8C2, (C6×Q8).64C22, C2.15(C123D4), (C2×C12).404C23, C42.S314C2, (C4×C12).133C22, C6.96(C8.C22), (C2×D12).108C22, C2.17(Q8.11D6), (C2×Dic6).113C22, (C3×C4⋊Q8)⋊4C2, (C2×C3⋊Q16)⋊15C2, (C2×C6).535(C2×D4), (C2×C4).73(C3⋊D4), (C2×C3⋊C8).137C22, (C2×Q82S3).7C2, (C2×C4).501(C22×S3), C22.207(C2×C3⋊D4), SmallGroup(192,645)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.80D6
C1C3C6C12C2×C12C2×D12C427S3 — C42.80D6
C3C6C2×C12 — C42.80D6
C1C22C42C4⋊Q8

Generators and relations for C42.80D6
 G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 368 in 124 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], S3, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8 [×2], C2×Q8, C3⋊C8 [×4], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×4], C22×S3, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C2×C3⋊C8 [×2], D6⋊C4 [×2], Q82S3 [×4], C3⋊Q16 [×4], C4×C12, C3×C4⋊C4 [×2], C2×Dic6, C2×D12, C6×Q8 [×2], C8.2D4, C42.S3, C427S3, C2×Q82S3 [×2], C2×C3⋊Q16 [×2], C3×C4⋊Q8, C42.80D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C8.C22 [×2], S3×D4 [×2], C2×C3⋊D4, C8.2D4, C123D4, Q8.11D6 [×2], C42.80D6

Character table of C42.80D6

 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
ρ102-2-22022-200000-2-2220-200-220000000    orthogonal lifted from D4
ρ1122220-122-2-22-20-1-1-100001-1-11111-1-11    orthogonal lifted from D6
ρ12222202-2-2-2200022200002-2-2-2-220000    orthogonal lifted from D4
ρ132-2-2202-2200000-2-22020-202-20000000    orthogonal lifted from D4
ρ1422220-12222-2-20-1-1-10000-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1522220-12222220-1-1-10000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ162-2-2202-2200000-2-220-20202-20000000    orthogonal lifted from D4
ρ1722220-122-2-2-220-1-1-100001-1-1111-111-1    orthogonal lifted from D6
ρ182-2-22022-200000-2-22-20200-220000000    orthogonal lifted from D4
ρ1922220-1-2-22-2000-1-1-10000111-1-11--3--3-3-3    complex lifted from C3⋊D4
ρ2022220-1-2-2-22000-1-1-10000-11111-1--3-3--3-3    complex lifted from C3⋊D4
ρ2122220-1-2-2-22000-1-1-10000-11111-1-3--3-3--3    complex lifted from C3⋊D4
ρ2222220-1-2-22-2000-1-1-10000111-1-11-3-3--3--3    complex lifted from C3⋊D4
ρ234-4-440-24-40000022-2000002-20000000    orthogonal lifted from S3×D4
ρ244-4-440-2-440000022-200000-220000000    orthogonal lifted from S3×D4
ρ254-44-40400000004-4-400000000000000    symplectic lifted from C8.C22, 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 Q8.11D6
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 Q8.11D6

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

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

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

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

Matrix representation of C42.80D6 in GL6(𝔽73)

010000
7200000
0030606047
0013432613
0043134313
0060306030
,
7200000
0720000
00720710
00072071
001010
000101
,
0720000
7200000
006944755
0069651865
006460469
0013448
,
0720000
100000
0022221826
00051855
006460469
006996569

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,30,13,43,60,0,0,60,43,13,30,0,0,60,26,43,60,0,0,47,13,13,30],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,69,69,64,13,0,0,4,65,60,4,0,0,47,18,4,4,0,0,55,65,69,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,22,0,64,69,0,0,22,51,60,9,0,0,18,8,4,65,0,0,26,55,69,69] >;

C42.80D6 in GAP, Magma, Sage, TeX

C_4^2._{80}D_6
% in TeX

G:=Group("C4^2.80D6");
// GroupNames label

G:=SmallGroup(192,645);
// by ID

G=gap.SmallGroup(192,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^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 C42.80D6 in TeX

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