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## G = C3×C42.29C22order 192 = 26·3

### Direct product of C3 and C42.29C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C42.29C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×D4 — C3×D4⋊C4 — C3×C42.29C22
 Lower central C1 — C2 — C2×C4 — C3×C42.29C22
 Upper central C1 — C2×C6 — C4×C12 — C3×C42.29C22

Generators and relations for C3×C42.29C22
G = < a,b,c,d,e | a3=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 242 in 110 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, D4⋊C4, C42.C2, C41D4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×D4, C6×D4, C42.29C22, C3×C8⋊C4, C3×D4⋊C4, C3×C42.C2, C3×C41D4, C3×C42.29C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C8⋊C22, C6×D4, C3×C4○D4, C42.29C22, C3×C4.4D4, C3×C8⋊C22, C3×C42.29C22

Smallest permutation representation of C3×C42.29C22
On 96 points
Generators in S96
(1 67 19)(2 68 20)(3 69 21)(4 70 22)(5 71 23)(6 72 24)(7 65 17)(8 66 18)(9 30 58)(10 31 59)(11 32 60)(12 25 61)(13 26 62)(14 27 63)(15 28 64)(16 29 57)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 52 89)(42 53 90)(43 54 91)(44 55 92)(45 56 93)(46 49 94)(47 50 95)(48 51 96)
(1 44 77 26)(2 41 78 31)(3 46 79 28)(4 43 80 25)(5 48 73 30)(6 45 74 27)(7 42 75 32)(8 47 76 29)(9 23 96 33)(10 20 89 38)(11 17 90 35)(12 22 91 40)(13 19 92 37)(14 24 93 34)(15 21 94 39)(16 18 95 36)(49 87 64 69)(50 84 57 66)(51 81 58 71)(52 86 59 68)(53 83 60 65)(54 88 61 70)(55 85 62 67)(56 82 63 72)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)
(1 26)(2 47)(3 32)(4 45)(5 30)(6 43)(7 28)(8 41)(9 23)(10 36)(11 21)(12 34)(13 19)(14 40)(15 17)(16 38)(18 89)(20 95)(22 93)(24 91)(25 74)(27 80)(29 78)(31 76)(33 96)(35 94)(37 92)(39 90)(42 79)(44 77)(46 75)(48 73)(49 83)(50 68)(51 81)(52 66)(53 87)(54 72)(55 85)(56 70)(57 86)(58 71)(59 84)(60 69)(61 82)(62 67)(63 88)(64 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)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 8 8 1 1 2 2 4 4 8 8 1 ··· 1 8 8 8 8 4 4 4 4 2 2 2 2 4 4 4 4 8 8 8 8 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 C8⋊C22 C3×C8⋊C22 kernel C3×C42.29C22 C3×C8⋊C4 C3×D4⋊C4 C3×C42.C2 C3×C4⋊1D4 C42.29C22 C8⋊C4 D4⋊C4 C42.C2 C4⋊1D4 C2×C12 C12 C2×C4 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 4 4 8 2 4

Matrix representation of C3×C42.29C22 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 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 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 41 35 21 21 0 0 0 0 38 44 0 52 0 0 0 0 47 47 67 0 0 0 0 0 53 6 38 67
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 52 52 1 2 0 0 0 0 21 0 72 72
,
 0 27 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 52 72 0 0 0 0 0 0 0 0 32 38 52 52 0 0 0 0 38 44 0 52 0 0 0 0 46 67 6 12 0 0 0 0 68 1 35 64
,
 27 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 52 71 0 0 0 0 0 0 2 21 0 0 0 0 0 0 0 0 46 67 6 12 0 0 0 0 26 26 6 0 0 0 0 0 41 35 21 21 0 0 0 0 23 61 6 53

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,41,38,47,53,0,0,0,0,35,44,47,6,0,0,0,0,21,0,67,38,0,0,0,0,21,52,0,67],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,52,21,0,0,0,0,72,0,52,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,1,52,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,32,38,46,68,0,0,0,0,38,44,67,1,0,0,0,0,52,0,6,35,0,0,0,0,52,52,12,64],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,52,2,0,0,0,0,0,0,71,21,0,0,0,0,0,0,0,0,46,26,41,23,0,0,0,0,67,26,35,61,0,0,0,0,6,6,21,6,0,0,0,0,12,0,21,53] >;

C3×C42.29C22 in GAP, Magma, Sage, TeX

C_3\times C_4^2._{29}C_2^2
% in TeX

G:=Group("C3xC4^2.29C2^2");
// GroupNames label

G:=SmallGroup(192,923);
// by ID

G=gap.SmallGroup(192,923);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1016,1094,1059,142,4204,172,6053,124]);
// Polycyclic

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

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