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G = C42.236D6order 192 = 26·3

56th non-split extension by C42 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.236D6, (C4xS3):3Q8, C12:Q8:34C2, D6.3(C2xQ8), C4.38(S3xQ8), C4:C4.204D6, C12.49(C2xQ8), (S3xC42).8C2, D6:Q8.1C2, C42.C2:15S3, C6.41(C22xQ8), (C2xC12).86C23, (C2xC6).232C24, D6:C4.38C22, C12.6Q8:22C2, Dic3.16(C2xQ8), Dic6:C4:34C2, (C4xC12).192C22, Dic3.12(C4oD4), Dic3:C4.50C22, C4:Dic3.239C22, C22.253(S3xC23), (C22xS3).219C23, C3:5(C23.37C23), (C4xDic3).139C22, (C2xDic3).311C23, (C2xDic6).178C22, C2.24(C2xS3xQ8), C2.84(S3xC4oD4), C6.195(C2xC4oD4), C4:C4:7S3.11C2, (C3xC42.C2):5C2, (S3xC2xC4).249C22, (C2xC4).77(C22xS3), (C3xC4:C4).187C22, SmallGroup(192,1247)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C42.236D6
C1C3C6C2xC6C22xS3S3xC2xC4S3xC42 — C42.236D6
C3C2xC6 — C42.236D6
C1C22C42.C2

Generators and relations for C42.236D6
 G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c5 >

Subgroups: 480 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xQ8, Dic6, C4xS3, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C2xC42, C42:C2, C4xQ8, C22:Q8, C42.C2, C42.C2, C4:Q8, C4xDic3, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, C2xDic6, S3xC2xC4, S3xC2xC4, C23.37C23, C12.6Q8, S3xC42, Dic6:C4, C12:Q8, C4:C4:7S3, D6:Q8, C3xC42.C2, C42.236D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C4oD4, C24, C22xS3, C22xQ8, C2xC4oD4, S3xQ8, S3xC23, C23.37C23, C2xS3xQ8, S3xC4oD4, C42.236D6

Smallest permutation representation of C42.236D6
On 96 points
Generators in S96
(1 54 35 76)(2 77 36 55)(3 56 25 78)(4 79 26 57)(5 58 27 80)(6 81 28 59)(7 60 29 82)(8 83 30 49)(9 50 31 84)(10 73 32 51)(11 52 33 74)(12 75 34 53)(13 85 71 47)(14 48 72 86)(15 87 61 37)(16 38 62 88)(17 89 63 39)(18 40 64 90)(19 91 65 41)(20 42 66 92)(21 93 67 43)(22 44 68 94)(23 95 69 45)(24 46 70 96)
(1 23 7 17)(2 64 8 70)(3 13 9 19)(4 66 10 72)(5 15 11 21)(6 68 12 62)(14 26 20 32)(16 28 22 34)(18 30 24 36)(25 71 31 65)(27 61 33 67)(29 63 35 69)(37 74 43 80)(38 59 44 53)(39 76 45 82)(40 49 46 55)(41 78 47 84)(42 51 48 57)(50 91 56 85)(52 93 58 87)(54 95 60 89)(73 86 79 92)(75 88 81 94)(77 90 83 96)
(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 6 29 34)(2 33 30 5)(3 4 31 32)(7 12 35 28)(8 27 36 11)(9 10 25 26)(13 14 65 66)(15 24 67 64)(16 63 68 23)(17 22 69 62)(18 61 70 21)(19 20 71 72)(37 46 93 90)(38 89 94 45)(39 44 95 88)(40 87 96 43)(41 42 85 86)(47 48 91 92)(49 80 77 52)(50 51 78 79)(53 76 81 60)(54 59 82 75)(55 74 83 58)(56 57 84 73)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T4U4V6A6B6C12A···12F12G12H12I12J
order12222234···4444444444444444466612···1212121212
size11116622···2333344446666121212122224···48888

42 irreducible representations

dim111111112222244
type+++++++++-++-
imageC1C2C2C2C2C2C2C2S3Q8D6D6C4oD4S3xQ8S3xC4oD4
kernelC42.236D6C12.6Q8S3xC42Dic6:C4C12:Q8C4:C4:7S3D6:Q8C3xC42.C2C42.C2C4xS3C42C4:C4Dic3C4C2
# reps111422411416824

Matrix representation of C42.236D6 in GL6(F13)

800000
050000
001000
000100
0000120
0000012
,
1200000
010000
001000
000100
000039
0000910
,
0120000
1200000
00121200
001000
000001
0000120
,
010000
1200000
00121200
000100
000001
0000120

G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,9,0,0,0,0,9,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C42.236D6 in GAP, Magma, Sage, TeX

C_4^2._{236}D_6
% in TeX

G:=Group("C4^2.236D6");
// GroupNames label

G:=SmallGroup(192,1247);
// by ID

G=gap.SmallGroup(192,1247);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,570,409,80,6278]);
// Polycyclic

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

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