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G = C42.65D6order 192 = 26·3

65th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.65D6, C3⋊C8.7D4, C4.13(S3×D4), (C2×D4).50D6, (C2×C12).83D4, C12.27(C2×D4), C32(C8.2D4), (C2×Q8).64D6, C122Q818C2, C4.4D4.7S3, C6.19(C41D4), (C6×D4).66C22, (C6×Q8).58C22, C2.10(C123D4), (C4×C12).109C22, (C2×C12).378C23, C42.S310C2, C2.19(Q8.14D6), C6.120(C8.C22), (C2×Dic6).108C22, (C2×C3⋊Q16)⋊14C2, (C2×C6).509(C2×D4), (C2×D4.S3).7C2, (C2×C4).63(C3⋊D4), (C2×C3⋊C8).123C22, (C3×C4.4D4).5C2, (C2×C4).478(C22×S3), C22.184(C2×C3⋊D4), SmallGroup(192,619)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.65D6
C1C3C6C12C2×C12C2×Dic6C122Q8 — C42.65D6
C3C6C2×C12 — C42.65D6
C1C22C42C4.4D4

Generators and relations for C42.65D6
 G = < a,b,c,d | a4=b4=c6=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 336 in 124 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8, C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], D4.S3 [×4], C3⋊Q16 [×4], C4×C12, C3×C22⋊C4 [×2], C2×Dic6 [×2], C6×D4, C6×Q8, C8.2D4, C42.S3, C122Q8, C2×D4.S3 [×2], C2×C3⋊Q16 [×2], C3×C4.4D4, C42.65D6
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, Q8.14D6 [×2], C42.65D6

Character table of C42.65D6

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
 size 111182224482424222881212121244444488
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-111111-1-1-1-111111111    linear of order 2
ρ311111111-1-1-1-1111111-11-111-1-11-1-1-1-1    linear of order 2
ρ411111111-1-1-11-1111111-11-11-1-11-1-1-1-1    linear of order 2
ρ51111-1111-1-111-1111-1-1-11-111-1-11-1-111    linear of order 2
ρ61111-1111-1-11-11111-1-11-11-11-1-11-1-111    linear of order 2
ρ71111-111111-1-1-1111-1-11111111111-1-1    linear of order 2
ρ81111-111111-111111-1-1-1-1-1-1111111-1-1    linear of order 2
ρ9222202-2-22-2000222000000-222-2-2-200    orthogonal lifted from D4
ρ102-2-2202-2200000-22-2000-202-20020000    orthogonal lifted from D4
ρ112222-2-12222-200-1-1-1110000-1-1-1-1-1-111    orthogonal lifted from D6
ρ1222222-12222200-1-1-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-2-22022-200000-22-200-2020200-20000    orthogonal lifted from D4
ρ142222-2-122-2-2200-1-1-1110000-111-111-1-1    orthogonal lifted from D6
ρ152-2-22022-200000-22-20020-20200-20000    orthogonal lifted from D4
ρ1622222-122-2-2-200-1-1-1-1-10000-111-11111    orthogonal lifted from D6
ρ172-2-2202-2200000-22-200020-2-20020000    orthogonal lifted from D4
ρ18222202-2-2-22000222000000-2-2-2-22200    orthogonal lifted from D4
ρ1922220-1-2-22-2000-1-1-1-3--300001-1-1111--3-3    complex lifted from C3⋊D4
ρ2022220-1-2-2-22000-1-1-1-3--300001111-1-1-3--3    complex lifted from C3⋊D4
ρ2122220-1-2-2-22000-1-1-1--3-300001111-1-1--3-3    complex lifted from C3⋊D4
ρ2222220-1-2-22-2000-1-1-1--3-300001-1-1111-3--3    complex lifted from C3⋊D4
ρ234-4-440-24-4000002-22000000-20020000    orthogonal lifted from S3×D4
ρ244-4-440-2-44000002-22000000200-20000    orthogonal lifted from S3×D4
ρ254-44-40400000004-4-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-4040000000-4-4400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2744-4-40-2000000022-2000000023-2300000    symplectic lifted from Q8.14D6, Schur index 2
ρ2844-4-40-2000000022-20000000-232300000    symplectic lifted from Q8.14D6, Schur index 2
ρ294-44-40-20000000-222000000000023-2300    symplectic lifted from Q8.14D6, Schur index 2
ρ304-44-40-20000000-2220000000000-232300    symplectic lifted from Q8.14D6, Schur index 2

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

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

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

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

Matrix representation of C42.65D6 in GL8(𝔽73)

15286900000
45600690000
431458450000
592928130000
0000006659
000000147
000071400
0000596600
,
10000000
01000000
00100000
00010000
00000010
00000001
000072000
000007200
,
7272000000
10000000
4329110000
44147200000
0000727200
00001000
00000011
000000720
,
4245000000
331000000
441942450000
48293310000
00006928445
00003244169
000069286928
0000324324

G:=sub<GL(8,GF(73))| [15,45,43,59,0,0,0,0,28,60,14,29,0,0,0,0,69,0,58,28,0,0,0,0,0,69,45,13,0,0,0,0,0,0,0,0,0,0,7,59,0,0,0,0,0,0,14,66,0,0,0,0,66,14,0,0,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,1,43,44,0,0,0,0,72,0,29,14,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0],[42,3,44,48,0,0,0,0,45,31,19,29,0,0,0,0,0,0,42,3,0,0,0,0,0,0,45,31,0,0,0,0,0,0,0,0,69,32,69,32,0,0,0,0,28,4,28,4,0,0,0,0,4,41,69,32,0,0,0,0,45,69,28,4] >;

C42.65D6 in GAP, Magma, Sage, TeX

C_4^2._{65}D_6
% in TeX

G:=Group("C4^2.65D6");
// GroupNames label

G:=SmallGroup(192,619);
// by ID

G=gap.SmallGroup(192,619);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,555,1123,297,136,6278]);
// Polycyclic

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

Export

Character table of C42.65D6 in TeX

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