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G = C42.155D6order 192 = 26·3

155th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.155D6, C6.1342+ 1+4, C4:C4.113D6, C12:D4:34C2, C4:D12:15C2, C42.C2:11S3, Dic3:5D4:38C2, C42:2S3:22C2, D6.D4:36C2, C2.59(D4oD12), (C2xC6).241C24, C12.131(C4oD4), (C4xC12).200C22, (C2xC12).189C23, D6:C4.112C22, C4.20(Q8:3S3), (C2xD12).166C22, C22.262(S3xC23), Dic3:C4.124C22, (C22xS3).106C23, C3:6(C22.34C24), (C4xDic3).146C22, (C2xDic3).261C23, C6.118(C2xC4oD4), (S3xC2xC4).131C22, C2.25(C2xQ8:3S3), (C3xC42.C2):14C2, (C3xC4:C4).196C22, (C2xC4).594(C22xS3), SmallGroup(192,1256)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C42.155D6
C1C3C6C2xC6C22xS3S3xC2xC4D6.D4 — C42.155D6
C3C2xC6 — C42.155D6
C1C22C42.C2

Generators and relations for C42.155D6
 G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c5 >

Subgroups: 736 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, C2xC6, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xS3, C42:C2, C4xD4, C4:D4, C22.D4, C42.C2, C4:1D4, C4xDic3, Dic3:C4, D6:C4, C4xC12, C3xC4:C4, S3xC2xC4, S3xC2xC4, C2xD12, C22.34C24, C42:2S3, C4:D12, Dic3:5D4, D6.D4, C12:D4, C3xC42.C2, C42.155D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, Q8:3S3, S3xC23, C22.34C24, C2xQ8:3S3, D4oD12, C42.155D6

Smallest permutation representation of C42.155D6
On 96 points
Generators in S96
(1 68 7 62)(2 76 8 82)(3 70 9 64)(4 78 10 84)(5 72 11 66)(6 80 12 74)(13 32 19 26)(14 51 20 57)(15 34 21 28)(16 53 22 59)(17 36 23 30)(18 55 24 49)(25 92 31 86)(27 94 33 88)(29 96 35 90)(37 63 43 69)(38 83 44 77)(39 65 45 71)(40 73 46 79)(41 67 47 61)(42 75 48 81)(50 87 56 93)(52 89 58 95)(54 91 60 85)
(1 34 42 52)(2 29 43 59)(3 36 44 54)(4 31 45 49)(5 26 46 56)(6 33 47 51)(7 28 48 58)(8 35 37 53)(9 30 38 60)(10 25 39 55)(11 32 40 50)(12 27 41 57)(13 79 93 72)(14 74 94 67)(15 81 95 62)(16 76 96 69)(17 83 85 64)(18 78 86 71)(19 73 87 66)(20 80 88 61)(21 75 89 68)(22 82 90 63)(23 77 91 70)(24 84 92 65)
(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 67 7 61)(2 72 8 66)(3 65 9 71)(4 70 10 64)(5 63 11 69)(6 68 12 62)(13 59 19 53)(14 52 20 58)(15 57 21 51)(16 50 22 56)(17 55 23 49)(18 60 24 54)(25 91 31 85)(26 96 32 90)(27 89 33 95)(28 94 34 88)(29 87 35 93)(30 92 36 86)(37 73 43 79)(38 78 44 84)(39 83 45 77)(40 76 46 82)(41 81 47 75)(42 74 48 80)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D···2H 3 4A4B4C···4H4I4J4K4L4M6A6B6C12A···12F12G12H12I12J
order12222···23444···44444466612···1212121212
size111112···122224···46666122224···48888

36 irreducible representations

dim11111112222444
type+++++++++++++
imageC1C2C2C2C2C2C2S3D6D6C4oD42+ 1+4Q8:3S3D4oD12
kernelC42.155D6C42:2S3C4:D12Dic3:5D4D6.D4C12:D4C3xC42.C2C42.C2C42C4:C4C12C6C4C2
# reps11124611164224

Matrix representation of C42.155D6 in GL8(F13)

012000000
10000000
00100000
00010000
00000120
000012002
000010012
00000110
,
01000000
120000000
001200000
000120000
00000100
000012000
00000001
000000120
,
08000000
80000000
00110000
001200000
00000500
00008000
00005008
00000550
,
80000000
08000000
00110000
000120000
00005003
00000830
00000550
00005008

G:=sub<GL(8,GF(13))| [0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,2,12,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,8,5,0,0,0,0,0,0,3,5,0,0,0,0,0,3,0,0,8] >;

C42.155D6 in GAP, Magma, Sage, TeX

C_4^2._{155}D_6
% in TeX

G:=Group("C4^2.155D6");
// GroupNames label

G:=SmallGroup(192,1256);
// by ID

G=gap.SmallGroup(192,1256);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,675,570,80,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations

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