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G = C3xC8.17D4order 192 = 26·3

Direct product of C3 and C8.17D4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C3xC8.17D4, C24.91D4, C12.63D8, Q16.2C12, M5(2).3C6, C8.3(C2xC12), C8.17(C3xD4), C4.12(C3xD8), C24.38(C2xC4), (C3xQ16).4C4, (C2xQ16).5C6, C8.C4.1C6, (C2xC12).281D4, (C6xQ16).12C2, (C2xC6).25SD16, (C3xM5(2)).7C2, C6.42(D4:C4), C22.4(C3xSD16), C12.74(C22:C4), (C2xC24).195C22, (C2xC8).14(C2xC6), (C2xC4).12(C3xD4), C4.6(C3xC22:C4), (C3xC8.C4).4C2, C2.11(C3xD4:C4), SmallGroup(192,168)

Series: Derived Chief Lower central Upper central

C1C8 — C3xC8.17D4
C1C2C4C2xC4C2xC8C2xC24C3xC8.C4 — C3xC8.17D4
C1C2C4C8 — C3xC8.17D4
C1C6C2xC12C2xC24 — C3xC8.17D4

Generators and relations for C3xC8.17D4
 G = < a,b,c,d | a3=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 98 in 54 conjugacy classes, 30 normal (26 characteristic)
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C12, C2xC6, C22:C4, D8, SD16, C2xC12, C3xD4, D4:C4, C3xC22:C4, C3xD8, C3xSD16, C8.17D4, C3xD4:C4, C3xC8.17D4
2C2
4C4
4C4
2C6
2Q8
2Q8
4C2xC4
4Q8
4C8
4C12
4C12
2C2xQ8
2M4(2)
2C16
2Q16
2C3xQ8
2C3xQ8
4C24
4C3xQ8
4C2xC12
2C6xQ8
2C3xQ16
2C48
2C3xM4(2)

Smallest permutation representation of C3xC8.17D4
On 96 points
Generators in S96
(1 96 45)(2 81 46)(3 82 47)(4 83 48)(5 84 33)(6 85 34)(7 86 35)(8 87 36)(9 88 37)(10 89 38)(11 90 39)(12 91 40)(13 92 41)(14 93 42)(15 94 43)(16 95 44)(17 61 66)(18 62 67)(19 63 68)(20 64 69)(21 49 70)(22 50 71)(23 51 72)(24 52 73)(25 53 74)(26 54 75)(27 55 76)(28 56 77)(29 57 78)(30 58 79)(31 59 80)(32 60 65)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)(65 79 77 75 73 71 69 67)(66 80 78 76 74 72 70 68)(81 95 93 91 89 87 85 83)(82 96 94 92 90 88 86 84)
(1 63 13 51 9 55 5 59)(2 50 6 62 10 58 14 54)(3 61 15 49 11 53 7 57)(4 64 8 60 12 56 16 52)(17 43 21 39 25 35 29 47)(18 38 30 42 26 46 22 34)(19 41 23 37 27 33 31 45)(20 36 32 40 28 44 24 48)(65 91 77 95 73 83 69 87)(66 94 70 90 74 86 78 82)(67 89 79 93 75 81 71 85)(68 92 72 88 76 84 80 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)

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

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

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

48 conjugacy classes

class 1 2A2B3A3B4A4B4C4D6A6B6C6D8A8B8C8D8E12A12B12C12D12E12F12G12H16A16B16C16D24A24B24C24D24E24F24G24H24I24J48A···48H
order1223344446666888881212121212121212161616162424242424242424242448···48
size11211228811222248822228888444422224488884···4

48 irreducible representations

dim11111111112222222244
type+++++++-
imageC1C2C2C2C3C4C6C6C6C12D4D4D8SD16C3xD4C3xD4C3xD8C3xSD16C8.17D4C3xC8.17D4
kernelC3xC8.17D4C3xC8.C4C3xM5(2)C6xQ16C8.17D4C3xQ16C8.C4M5(2)C2xQ16Q16C24C2xC12C12C2xC6C8C2xC4C4C22C3C1
# reps11112422281122224424

Matrix representation of C3xC8.17D4 in GL4(F7) generated by

4000
0400
0040
0004
,
5026
6556
6112
2262
,
1150
4533
6632
6305
,
4516
4143
2306
5052
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,4,6,6,1,5,6,3,5,3,3,0,0,3,2,5],[4,4,2,5,5,1,3,0,1,4,0,5,6,3,6,2] >;

C3xC8.17D4 in GAP, Magma, Sage, TeX

C_3\times C_8._{17}D_4
% in TeX

G:=Group("C3xC8.17D4");
// GroupNames label

G:=SmallGroup(192,168);
// by ID

G=gap.SmallGroup(192,168);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,2194,136,2111,172,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

Export

Subgroup lattice of C3xC8.17D4 in TeX

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