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G = C42.233D6order 192 = 26·3

53rd non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.233D6, (C4xS3):6D4, D6.6(C2xD4), C4.30(S3xD4), (S3xC42):8C2, C12.59(C2xD4), Dic3:D4:37C2, C12:3D4:23C2, (C2xD4).168D6, C4.4D4:18S3, (C2xQ8).158D6, C22:C4.70D6, C6.86(C22xD4), C42:7S3:22C2, Dic3:3(C4oD4), (C2xC12).76C23, (C2xC6).212C24, D6:C4.58C22, Dic3.65(C2xD4), Dic3:4D4:27C2, Dic3:Q8:19C2, (C4xC12).182C22, (C6xD4).150C22, (C22xC6).42C23, C23.44(C22xS3), (C6xQ8).121C22, (C2xD12).160C22, Dic3:C4.47C22, C22.233(S3xC23), C3:4(C22.26C24), (C22xS3).212C23, (C2xDic6).173C22, (C4xDic3).296C22, (C2xDic3).249C23, (C22xDic3).137C22, C2.59(C2xS3xD4), C2.71(S3xC4oD4), (C2xQ8:3S3):9C2, C6.183(C2xC4oD4), (C3xC4.4D4):6C2, (C2xD4:2S3):18C2, (S3xC2xC4).118C22, (C2xC4).298(C22xS3), (C2xC3:D4).55C22, (C3xC22:C4).59C22, SmallGroup(192,1227)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C42.233D6
C1C3C6C2xC6C2xDic3S3xC2xC4S3xC42 — C42.233D6
C3C2xC6 — C42.233D6
C1C22C4.4D4

Generators and relations for C42.233D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c-1 >

Subgroups: 816 in 310 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic6, C4xS3, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C2xC42, C4xD4, C4:D4, C4.4D4, C4.4D4, C4:1D4, C4:Q8, C2xC4oD4, C4xDic3, C4xDic3, Dic3:C4, D6:C4, C4xC12, C3xC22:C4, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, D4:2S3, Q8:3S3, C22xDic3, C2xC3:D4, C6xD4, C6xQ8, C22.26C24, S3xC42, C42:7S3, Dic3:4D4, Dic3:D4, C12:3D4, Dic3:Q8, C3xC4.4D4, C2xD4:2S3, C2xQ8:3S3, C42.233D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, S3xD4, S3xC23, C22.26C24, C2xS3xD4, S3xC4oD4, C42.233D6

Smallest permutation representation of C42.233D6
On 96 points
Generators in S96
(1 80 19 66)(2 68 20 53)(3 82 21 62)(4 70 22 49)(5 84 23 64)(6 72 24 51)(7 79 25 65)(8 67 26 52)(9 81 27 61)(10 69 28 54)(11 83 29 63)(12 71 30 50)(13 47 86 38)(14 58 87 34)(15 43 88 40)(16 60 89 36)(17 45 90 42)(18 56 85 32)(31 93 55 73)(33 95 57 75)(35 91 59 77)(37 94 46 74)(39 96 48 76)(41 92 44 78)
(1 59 8 43)(2 36 9 41)(3 55 10 45)(4 32 11 37)(5 57 12 47)(6 34 7 39)(13 64 95 50)(14 79 96 72)(15 66 91 52)(16 81 92 68)(17 62 93 54)(18 83 94 70)(19 35 26 40)(20 60 27 44)(21 31 28 42)(22 56 29 46)(23 33 30 38)(24 58 25 48)(49 85 63 74)(51 87 65 76)(53 89 61 78)(67 88 80 77)(69 90 82 73)(71 86 84 75)
(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 73)(2 92)(3 77)(4 96)(5 75)(6 94)(7 18)(8 90)(9 16)(10 88)(11 14)(12 86)(13 30)(15 28)(17 26)(19 93)(20 78)(21 91)(22 76)(23 95)(24 74)(25 85)(27 89)(29 87)(31 67)(32 51)(33 71)(34 49)(35 69)(36 53)(37 65)(38 84)(39 63)(40 82)(41 61)(42 80)(43 62)(44 81)(45 66)(46 79)(47 64)(48 83)(50 57)(52 55)(54 59)(56 72)(58 70)(60 68)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4F4G4H4I4J4K4L4M4N4O4P4Q4R6A6B6C6D6E12A···12F12G12H
order122222222234···44444444444446666612···121212
size11114466121222···233334466661212222884···488

42 irreducible representations

dim1111111111222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D4D6D6D6D6C4oD4S3xD4S3xC4oD4
kernelC42.233D6S3xC42C42:7S3Dic3:4D4Dic3:D4C12:3D4Dic3:Q8C3xC4.4D4C2xD4:2S3C2xQ8:3S3C4.4D4C4xS3C42C22:C4C2xD4C2xQ8Dic3C4C2
# reps1114411111141411824

Matrix representation of C42.233D6 in GL6(F13)

1200000
0120000
0001200
001000
000051
000028
,
100000
010000
000100
0012000
000080
000008
,
010000
1210000
0001200
0012000
000018
0000012
,
1210000
010000
0012000
000100
0000125
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,5,2,0,0,0,0,1,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,8,12],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,5,1] >;

C42.233D6 in GAP, Magma, Sage, TeX

C_4^2._{233}D_6
% in TeX

G:=Group("C4^2.233D6");
// GroupNames label

G:=SmallGroup(192,1227);
// by ID

G=gap.SmallGroup(192,1227);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,232,100,1123,346,297,6278]);
// Polycyclic

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

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