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## G = C4×C4.Dic3order 192 = 26·3

### Direct product of C4 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C2×C4.Dic3 — C4×C4.Dic3
 Lower central C3 — C6 — C4×C4.Dic3
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4.Dic3
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 216 in 142 conjugacy classes, 95 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×C12, C4×C12, C22×C12, C22×C12, C4×M4(2), C4×C3⋊C8, C42.S3, C2×C4.Dic3, C2×C4×C12, C4×C4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×M4(2), C4.Dic3, C4×Dic3, S3×C2×C4, C22×Dic3, C4×M4(2), C2×C4.Dic3, C2×C4×Dic3, C4×C4.Dic3

Smallest permutation representation of C4×C4.Dic3
On 96 points
Generators in S96
(1 28 72 81)(2 29 61 82)(3 30 62 83)(4 31 63 84)(5 32 64 73)(6 33 65 74)(7 34 66 75)(8 35 67 76)(9 36 68 77)(10 25 69 78)(11 26 70 79)(12 27 71 80)(13 52 96 38)(14 53 85 39)(15 54 86 40)(16 55 87 41)(17 56 88 42)(18 57 89 43)(19 58 90 44)(20 59 91 45)(21 60 92 46)(22 49 93 47)(23 50 94 48)(24 51 95 37)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 88 91 94)(86 89 92 95)(87 90 93 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 85 10 94 7 91 4 88)(2 90 11 87 8 96 5 93)(3 95 12 92 9 89 6 86)(13 64 22 61 19 70 16 67)(14 69 23 66 20 63 17 72)(15 62 24 71 21 68 18 65)(25 48 34 45 31 42 28 39)(26 41 35 38 32 47 29 44)(27 46 36 43 33 40 30 37)(49 82 58 79 55 76 52 73)(50 75 59 84 56 81 53 78)(51 80 60 77 57 74 54 83)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4L 4M ··· 4R 6A ··· 6G 8A ··· 8P 12A ··· 12X order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C4 S3 Dic3 D6 Dic3 D6 M4(2) C4×S3 C4.Dic3 kernel C4×C4.Dic3 C4×C3⋊C8 C42.S3 C2×C4.Dic3 C2×C4×C12 C4.Dic3 C4×C12 C22×C12 C2×C42 C42 C42 C22×C4 C22×C4 C12 C2×C4 C4 # reps 1 2 2 2 1 16 4 4 1 2 2 2 1 8 8 16

Matrix representation of C4×C4.Dic3 in GL4(𝔽73) generated by

 46 0 0 0 0 46 0 0 0 0 1 0 0 0 0 1
,
 46 27 0 0 0 27 0 0 0 0 46 0 0 0 0 27
,
 46 0 0 0 0 46 0 0 0 0 24 0 0 0 0 3
,
 46 14 0 0 19 27 0 0 0 0 0 1 0 0 46 0
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,27,27,0,0,0,0,46,0,0,0,0,27],[46,0,0,0,0,46,0,0,0,0,24,0,0,0,0,3],[46,19,0,0,14,27,0,0,0,0,0,46,0,0,1,0] >;

C4×C4.Dic3 in GAP, Magma, Sage, TeX

C_4\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C4xC4.Dic3");
// GroupNames label

G:=SmallGroup(192,481);
// by ID

G=gap.SmallGroup(192,481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,100,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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