Copied to
clipboard

G = C42.48D6order 192 = 26·3

48th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.48D6, D4⋊S35C4, D44(C4×S3), (C4×D4)⋊2S3, (D4×C12)⋊2C2, C6.70(C4×D4), (C4×D12)⋊20C2, D1211(C2×C4), C34(D8⋊C4), C4⋊C4.242D6, (C2×D4).189D6, (C2×C12).254D4, C6.D829C2, C4.37(C4○D12), C12.49(C4○D4), D4⋊Dic311C2, C2.3(D4⋊D6), (C4×C12).85C22, C42.S35C2, C12.22(C22×C4), C12.Q832C2, C6.108(C8⋊C22), (C2×C12).336C23, C2.3(D126C22), (C6×D4).231C22, (C2×D12).237C22, C4⋊Dic3.327C22, C3⋊C88(C2×C4), C4.22(S3×C2×C4), (C3×D4)⋊9(C2×C4), (C2×D4⋊S3).4C2, C2.16(C4×C3⋊D4), (C2×C6).467(C2×D4), (C2×C3⋊C8).92C22, C22.76(C2×C3⋊D4), (C2×C4).217(C3⋊D4), (C3×C4⋊C4).273C22, (C2×C4).436(C22×S3), SmallGroup(192,573)

Series: Derived Chief Lower central Upper central

C1C12 — C42.48D6
C1C3C6C2×C6C2×C12C2×D12C2×D4⋊S3 — C42.48D6
C3C6C12 — C42.48D6
C1C22C42C4×D4

Generators and relations for C42.48D6
 G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 376 in 132 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×8], S3 [×2], C6 [×3], C6 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×4], C23 [×2], Dic3, C12 [×2], C12 [×3], D6 [×4], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4, C2×D4, C3⋊C8 [×2], C3⋊C8, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×S3, C22×C6, C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4, C4×D4, C2×D8, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×4], C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, D8⋊C4, C42.S3, C12.Q8, C6.D8, D4⋊Dic3, C4×D12, C2×D4⋊S3, D4×C12, C42.48D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C8⋊C22 [×2], S3×C2×C4, C4○D12, C2×C3⋊D4, D8⋊C4, C4×C3⋊D4, D126C22, D4⋊D6, C42.48D6

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G4H4I4J6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E···12L
order1222222234···44444666666688881212121212···12
size111144121222···244121222244441212121222224···4

42 irreducible representations

dim111111111222222222444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D4D6D6D6C4○D4C3⋊D4C4×S3C4○D12C8⋊C22D126C22D4⋊D6
kernelC42.48D6C42.S3C12.Q8C6.D8D4⋊Dic3C4×D12C2×D4⋊S3D4×C12D4⋊S3C4×D4C2×C12C42C4⋊C4C2×D4C12C2×C4D4C4C6C2C2
# reps111111118121112444222

Matrix representation of C42.48D6 in GL6(𝔽73)

4600000
0460000
00003060
00001343
00431300
00603000
,
100000
010000
000010
000001
0072000
0007200
,
110000
7200000
0021333321
0040615212
0033215240
0052123312
,
110000
0720000
0033215240
0061406121
0021333321
0012526140

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,30,13,0,0,0,0,60,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,21,40,33,52,0,0,33,61,21,12,0,0,33,52,52,33,0,0,21,12,40,12],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,33,61,21,12,0,0,21,40,33,52,0,0,52,61,33,61,0,0,40,21,21,40] >;

C42.48D6 in GAP, Magma, Sage, TeX

C_4^2._{48}D_6
% in TeX

G:=Group("C4^2.48D6");
// GroupNames label

G:=SmallGroup(192,573);
// by ID

G=gap.SmallGroup(192,573);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,1684,851,102,6278]);
// Polycyclic

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

׿
×
𝔽