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G = C42.213D6order 192 = 26·3

33rd non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.213D6, (C2xD4).46D6, (C2xQ8).60D6, (C2xC12).270D4, C4.4D4.6S3, C6.104(C4oD8), C12.67(C4oD4), Q8:2Dic3:21C2, C12.6Q8:12C2, (C6xD4).62C22, (C6xQ8).54C22, C4.21(D4:2S3), (C4xC12).105C22, (C2xC12).374C23, D4:Dic3.13C2, C6.42(C4.4D4), C2.23(Q8.13D6), C2.9(C23.12D6), C4:Dic3.151C22, C3:4(C42.78C22), (C4xC3:C8):11C2, (C2xC6).505(C2xD4), (C2xC3:C8).252C22, (C3xC4.4D4).4C2, (C2xC4).109(C3:D4), (C2xC4).474(C22xS3), C22.180(C2xC3:D4), SmallGroup(192,615)

Series: Derived Chief Lower central Upper central

C1C2xC12 — C42.213D6
C1C3C6C2xC6C2xC12C2xC3:C8C4xC3:C8 — C42.213D6
C3C6C2xC12 — C42.213D6
C1C22C42C4.4D4

Generators and relations for C42.213D6
 G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 240 in 96 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xD4, C2xQ8, C3:C8, C2xDic3, C2xC12, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C4xC8, D4:C4, Q8:C4, C4.4D4, C42.C2, C2xC3:C8, Dic3:C4, C4:Dic3, C4xC12, C3xC22:C4, C6xD4, C6xQ8, C42.78C22, C4xC3:C8, D4:Dic3, Q8:2Dic3, C12.6Q8, C3xC4.4D4, C42.213D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4.4D4, C4oD8, D4:2S3, C2xC3:D4, C42.78C22, C23.12D6, Q8.13D6, C42.213D6

Smallest permutation representation of C42.213D6
On 96 points
Generators in S96
(1 18 11 56)(2 16 12 60)(3 14 10 58)(4 17 9 55)(5 15 7 59)(6 13 8 57)(19 44 34 47)(20 65 35 62)(21 46 36 43)(22 61 31 64)(23 48 32 45)(24 63 33 66)(25 75 91 54)(26 80 92 68)(27 77 93 50)(28 82 94 70)(29 73 95 52)(30 84 96 72)(37 71 89 83)(38 53 90 74)(39 67 85 79)(40 49 86 76)(41 69 87 81)(42 51 88 78)
(1 21 7 24)(2 19 8 22)(3 23 9 20)(4 35 10 32)(5 33 11 36)(6 31 12 34)(13 64 60 47)(14 48 55 65)(15 66 56 43)(16 44 57 61)(17 62 58 45)(18 46 59 63)(25 42 39 28)(26 29 40 37)(27 38 41 30)(49 71 80 73)(50 74 81 72)(51 67 82 75)(52 76 83 68)(53 69 84 77)(54 78 79 70)(85 94 91 88)(86 89 92 95)(87 96 93 90)
(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 52 11 73)(2 54 12 75)(3 50 10 77)(4 69 9 81)(5 71 7 83)(6 67 8 79)(13 25 57 91)(14 87 58 41)(15 29 59 95)(16 85 60 39)(17 27 55 93)(18 89 56 37)(19 70 34 82)(20 74 35 53)(21 68 36 80)(22 78 31 51)(23 72 32 84)(24 76 33 49)(26 63 92 66)(28 61 94 64)(30 65 96 62)(38 48 90 45)(40 46 86 43)(42 44 88 47)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D 3 4A···4F4G4H4I6A6B6C6D6E8A···8H12A···12F12G12H
order1222234···4444666668···812···121212
size1111822···282424222886···64···488

36 irreducible representations

dim1111112222222244
type+++++++++++-
imageC1C2C2C2C2C2S3D4D6D6D6C4oD4C3:D4C4oD8D4:2S3Q8.13D6
kernelC42.213D6C4xC3:C8D4:Dic3Q8:2Dic3C12.6Q8C3xC4.4D4C4.4D4C2xC12C42C2xD4C2xQ8C12C2xC4C6C4C2
# reps1122111211144824

Matrix representation of C42.213D6 in GL6(F73)

4600000
0460000
0072000
0007200
00004630
00003927
,
010000
7200000
0072000
0007200
00007266
0000421
,
100000
0720000
0064000
00536500
000010
00003172
,
6760000
660000
00694400
0071400
0000034
0000580

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,39,0,0,0,0,30,27],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,42,0,0,0,0,66,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,64,53,0,0,0,0,0,65,0,0,0,0,0,0,1,31,0,0,0,0,0,72],[67,6,0,0,0,0,6,6,0,0,0,0,0,0,69,71,0,0,0,0,44,4,0,0,0,0,0,0,0,58,0,0,0,0,34,0] >;

C42.213D6 in GAP, Magma, Sage, TeX

C_4^2._{213}D_6
% in TeX

G:=Group("C4^2.213D6");
// GroupNames label

G:=SmallGroup(192,615);
// by ID

G=gap.SmallGroup(192,615);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,471,438,102,6278]);
// Polycyclic

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

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