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G = C42.70D6order 192 = 26·3

70th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.70D6, C4⋊C4.76D6, (C2×C12).85D4, C42.C22S3, C4⋊D12.7C2, C6.D841C2, C12.71(C4○D4), C2.22(D4⋊D6), C6.123(C8⋊C22), (C2×C12).385C23, C42.S311C2, (C4×C12).115C22, C4.13(Q83S3), C6.55(C4.4D4), C2.8(C12.23D4), (C2×D12).103C22, C33(C42.29C22), (C2×C6).516(C2×D4), (C3×C42.C2)⋊2C2, (C2×C4).67(C3⋊D4), (C2×C3⋊C8).127C22, (C3×C4⋊C4).123C22, (C2×C4).483(C22×S3), C22.189(C2×C3⋊D4), SmallGroup(192,626)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.70D6
C1C3C6C12C2×C12C2×D12C4⋊D12 — C42.70D6
C3C6C2×C12 — C42.70D6
C1C22C42C42.C2

Generators and relations for C42.70D6
 G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 416 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], C12 [×2], C12 [×4], D6 [×6], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×D4 [×4], C3⋊C8 [×2], D12 [×8], C2×C12, C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C2×C3⋊C8 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×D12 [×2], C2×D12 [×2], C42.29C22, C42.S3, C6.D8 [×4], C4⋊D12, C3×C42.C2, C42.70D6
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 [×2], Q83S3 [×2], C2×C3⋊D4, C42.29C22, C12.23D4, D4⋊D6 [×2], C42.70D6

Character table of C42.70D6

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H12I12J
 size 111124242224488222121212124444448888
ρ1111111111111111111111111111111    trivial
ρ21111-11111-1-11-1111-111-1-1-1-11-11-1-111    linear of order 2
ρ311111111111-1-1111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ41111-11111-1-1-111111-1-11-1-1-11-1111-1-1    linear of order 2
ρ511111-1111-1-11-11111-1-11-1-1-11-11-1-111    linear of order 2
ρ61111-1-11111111111-1-1-1-11111111111    linear of order 2
ρ711111-1111-1-1-11111-111-1-1-1-11-1111-1-1    linear of order 2
ρ81111-1-111111-1-11111111111111-1-1-1-1    linear of order 2
ρ9222200-122-2-2-22-1-1-10000111-11-1-1-111    orthogonal lifted from D6
ρ102222002-2-22-2002220000-22-2-22-20000    orthogonal lifted from D4
ρ11222200-122-2-22-2-1-1-10000111-11-111-1-1    orthogonal lifted from D6
ρ122222002-2-2-220022200002-22-2-2-20000    orthogonal lifted from D4
ρ13222200-1222222-1-1-10000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222200-12222-2-2-1-1-10000-1-1-1-1-1-11111    orthogonal lifted from D6
ρ15222200-1-2-2-2200-1-1-10000-11-1111--3-3--3-3    complex lifted from C3⋊D4
ρ16222200-1-2-22-200-1-1-100001-111-11--3-3-3--3    complex lifted from C3⋊D4
ρ17222200-1-2-2-2200-1-1-10000-11-1111-3--3-3--3    complex lifted from C3⋊D4
ρ18222200-1-2-22-200-1-1-100001-111-11-3--3--3-3    complex lifted from C3⋊D4
ρ1922-2-20022-20000-2-22-2i002i00020-20000    complex lifted from C4○D4
ρ2022-2-20022-20000-2-222i00-2i00020-20000    complex lifted from C4○D4
ρ2122-2-2002-220000-2-2202i-2i0000-2020000    complex lifted from C4○D4
ρ2222-2-2002-220000-2-220-2i2i0000-2020000    complex lifted from C4○D4
ρ2344-4-400-24-4000022-20000000-2020000    orthogonal lifted from Q83S3, Schur index 2
ρ244-4-440040000004-4-400000000000000    orthogonal lifted from C8⋊C22
ρ254-44-4004000000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ2644-4-400-2-44000022-2000000020-20000    orthogonal lifted from Q83S3, Schur index 2
ρ274-44-400-20000002-220000230-230000000    orthogonal lifted from D4⋊D6
ρ284-44-400-20000002-220000-230230000000    orthogonal lifted from D4⋊D6
ρ294-4-4400-2000000-222000002300-2300000    orthogonal lifted from D4⋊D6
ρ304-4-4400-2000000-22200000-23002300000    orthogonal lifted from D4⋊D6

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

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

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

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

Matrix representation of C42.70D6 in GL8(𝔽73)

474000000
6826000000
00100000
00010000
0000026658
000047088
0000865026
00006565470
,
10000000
01000000
00100000
00010000
00000100
000072000
000000072
00000010
,
270000000
5946000000
000720000
001720000
00000010
00000001
000072000
000007200
,
270000000
027000000
001720000
000720000
00000001
00000010
000072000
00000100

G:=sub<GL(8,GF(73))| [47,68,0,0,0,0,0,0,4,26,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,47,8,65,0,0,0,0,26,0,65,65,0,0,0,0,65,8,0,47,0,0,0,0,8,8,26,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[27,59,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C42.70D6 in GAP, Magma, Sage, TeX

C_4^2._{70}D_6
% in TeX

G:=Group("C4^2.70D6");
// GroupNames label

G:=SmallGroup(192,626);
// by ID

G=gap.SmallGroup(192,626);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,555,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,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 C42.70D6 in TeX

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