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## G = C2×D4×Dic3order 192 = 26·3

### Direct product of C2, D4 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D4×Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C2×D4×Dic3
 Lower central C3 — C6 — C2×D4×Dic3
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×D4×Dic3
G = < a,b,c,d,e | a2=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 840 in 426 conjugacy classes, 215 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×Dic3, C22×Dic3, C22×C12, C6×D4, C23×C6, C2×C4×D4, C2×C4×Dic3, C2×C4⋊Dic3, D4×Dic3, C2×C6.D4, C23×Dic3, D4×C2×C6, C2×D4×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C24, C2×Dic3, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×D4, D42S3, C22×Dic3, S3×C23, C2×C4×D4, D4×Dic3, C2×S3×D4, C2×D42S3, C23×Dic3, C2×D4×Dic3

Smallest permutation representation of C2×D4×Dic3
On 96 points
Generators in S96
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(55 90)(56 85)(57 86)(58 87)(59 88)(60 89)(61 80)(62 81)(63 82)(64 83)(65 84)(66 79)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(1 51 16 48)(2 52 17 43)(3 53 18 44)(4 54 13 45)(5 49 14 46)(6 50 15 47)(7 62 93 57)(8 63 94 58)(9 64 95 59)(10 65 96 60)(11 66 91 55)(12 61 92 56)(19 41 29 31)(20 42 30 32)(21 37 25 33)(22 38 26 34)(23 39 27 35)(24 40 28 36)(67 90 78 79)(68 85 73 80)(69 86 74 81)(70 87 75 82)(71 88 76 83)(72 89 77 84)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 77)(8 78)(9 73)(10 74)(11 75)(12 76)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 51)(26 52)(27 53)(28 54)(29 49)(30 50)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)
(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 81 4 84)(2 80 5 83)(3 79 6 82)(7 21 10 24)(8 20 11 23)(9 19 12 22)(13 89 16 86)(14 88 17 85)(15 87 18 90)(25 96 28 93)(26 95 29 92)(27 94 30 91)(31 56 34 59)(32 55 35 58)(33 60 36 57)(37 65 40 62)(38 64 41 61)(39 63 42 66)(43 73 46 76)(44 78 47 75)(45 77 48 74)(49 71 52 68)(50 70 53 67)(51 69 54 72)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A 4B 4C 4D 4E ··· 4L 4M ··· 4X 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 ··· 2 3 4 4 4 4 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 2 ··· 2 2 2 2 2 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4 4 4 4 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 D6 C4○D4 S3×D4 D4⋊2S3 kernel C2×D4×Dic3 C2×C4×Dic3 C2×C4⋊Dic3 D4×Dic3 C2×C6.D4 C23×Dic3 D4×C2×C6 C6×D4 C22×D4 C2×Dic3 C22×C4 C2×D4 C2×D4 C24 C2×C6 C22 C22 # reps 1 1 1 8 2 2 1 16 1 4 1 8 4 2 4 2 2

Matrix representation of C2×D4×Dic3 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 0
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 12 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 8 5 0 0 0 0 5 0 0 0 0 0 8 0 0 0 0 0 8

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,0],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,8,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,8] >;

C2×D4×Dic3 in GAP, Magma, Sage, TeX

C_2\times D_4\times {\rm Dic}_3
% in TeX

G:=Group("C2xD4xDic3");
// GroupNames label

G:=SmallGroup(192,1354);
// by ID

G=gap.SmallGroup(192,1354);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,297,6278]);
// Polycyclic

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

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