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## G = C3×C42.6C22order 192 = 26·3

### Direct product of C3 and C42.6C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C42.6C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C4⋊C8 — C3×C42.6C22
 Lower central C1 — C22 — C3×C42.6C22
 Upper central C1 — C2×C12 — C3×C42.6C22

Generators and relations for C3×C42.6C22
G = < a,b,c,d,e | a3=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >

Subgroups: 146 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C22×C12, C42.6C22, C3×C4⋊C8, C3×C42⋊C2, C22×C24, C6×M4(2), C3×C42.6C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C8○D4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C42.6C22, C6×C4⋊C4, C3×C8○D4, C3×C42.6C22

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C3 C4 C4 C6 C6 C6 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 C8○D4 C3×C8○D4 kernel C3×C42.6C22 C3×C4⋊C8 C3×C42⋊C2 C22×C24 C6×M4(2) C42.6C22 C3×C22⋊C4 C3×C4⋊C4 C4⋊C8 C42⋊C2 C22×C8 C2×M4(2) C22⋊C4 C4⋊C4 C2×C12 C2×C12 C2×C4 C2×C4 C6 C2 # reps 1 4 1 1 1 2 4 4 8 2 2 2 8 8 2 2 4 4 8 16

Matrix representation of C3×C42.6C22 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 28 68 0 0 11 45
,
 46 0 0 0 0 46 0 0 0 0 72 0 0 0 0 72
,
 22 0 0 0 0 51 0 0 0 0 41 42 0 0 26 32
,
 0 1 0 0 72 0 0 0 0 0 45 5 0 0 62 28
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,28,11,0,0,68,45],[46,0,0,0,0,46,0,0,0,0,72,0,0,0,0,72],[22,0,0,0,0,51,0,0,0,0,41,26,0,0,42,32],[0,72,0,0,1,0,0,0,0,0,45,62,0,0,5,28] >;

C3×C42.6C22 in GAP, Magma, Sage, TeX

C_3\times C_4^2._6C_2^2
% in TeX

G:=Group("C3xC4^2.6C2^2");
// GroupNames label

G:=SmallGroup(192,857);
// by ID

G=gap.SmallGroup(192,857);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,124]);
// Polycyclic

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

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