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G = C42.61D6order 192 = 26·3

61st non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.61D6, Dic6.23D4, C4.49(S3×D4), C12⋊C828C2, (C2×D4).44D6, C12.23(C2×D4), (C2×Q8).58D6, (C4×Dic6)⋊21C2, (C2×C12).269D4, C35(Q8.D4), C4.4D4.4S3, C6.103(C4○D8), C12.65(C4○D4), C4.1(D42S3), Q82Dic319C2, (C6×D4).60C22, (C6×Q8).52C22, C2.10(D63D4), C6.101(C4⋊D4), (C4×C12).103C22, (C2×C12).372C23, D4⋊Dic3.11C2, C2.17(Q8.14D6), C2.22(Q8.13D6), C6.118(C8.C22), C4⋊Dic3.340C22, (C2×Dic6).272C22, (C2×C3⋊Q16)⋊12C2, (C2×C6).503(C2×D4), (C2×D4.S3).6C2, (C2×C4).59(C3⋊D4), (C2×C3⋊C8).119C22, (C3×C4.4D4).2C2, (C2×C4).472(C22×S3), C22.178(C2×C3⋊D4), SmallGroup(192,613)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.61D6
C1C3C6C12C2×C12C4⋊Dic3C4×Dic6 — C42.61D6
C3C6C2×C12 — C42.61D6
C1C22C42C4.4D4

Generators and relations for C42.61D6
 G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 288 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×6], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, Dic3 [×3], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C22⋊C4 [×2], C2×Dic6, C6×D4, C6×Q8, Q8.D4, C12⋊C8, D4⋊Dic3, Q82Dic3, C4×Dic6, C2×D4.S3, C2×C3⋊Q16, C3×C4.4D4, C42.61D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8.C22, S3×D4, D42S3, C2×C3⋊D4, Q8.D4, D63D4, Q8.13D6, Q8.14D6, C42.61D6

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

33 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E8A8B8C8D12A···12F12G12H
order122223444444444466666888812···121212
size1111822222481212121222288121212124···488

33 irreducible representations

dim1111111122222222244444
type++++++++++++++-+--
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4C4○D8C8.C22S3×D4D42S3Q8.13D6Q8.14D6
kernelC42.61D6C12⋊C8D4⋊Dic3Q82Dic3C4×Dic6C2×D4.S3C2×C3⋊Q16C3×C4.4D4C4.4D4Dic6C2×C12C42C2×D4C2×Q8C12C2×C4C6C6C4C4C2C2
# reps1111111112211124411122

Matrix representation of C42.61D6 in GL6(𝔽73)

100000
010000
0046000
0004600
000001
0000720
,
100000
010000
0017100
0017200
0000720
0000072
,
72720000
100000
001000
0017200
000010
0000072
,
11710000
60620000
0003200
00573200
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[11,60,0,0,0,0,71,62,0,0,0,0,0,0,0,57,0,0,0,0,32,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C42.61D6 in GAP, Magma, Sage, TeX

C_4^2._{61}D_6
% in TeX

G:=Group("C4^2.61D6");
// GroupNames label

G:=SmallGroup(192,613);
// by ID

G=gap.SmallGroup(192,613);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,1123,297,136,6278]);
// Polycyclic

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

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