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## G = C2×C4×Dic3order 96 = 25·3

### Direct product of C2×C4 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C4×Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3
 Lower central C3 — C2×C4×Dic3
 Upper central C1 — C22×C4

Generators and relations for C2×C4×Dic3
G = < a,b,c,d | a2=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 162 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C2×Dic3, C2×C12, C22×C6, C2×C42, C4×Dic3, C22×Dic3, C22×C12, C2×C4×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C4×Dic3, S3×C2×C4, C22×Dic3, C2×C4×Dic3

Smallest permutation representation of C2×C4×Dic3
Regular action on 96 points
Generators in S96
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 77)(8 78)(9 73)(10 74)(11 75)(12 76)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(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 25 15 20)(2 26 16 21)(3 27 17 22)(4 28 18 23)(5 29 13 24)(6 30 14 19)(7 82 94 86)(8 83 95 87)(9 84 96 88)(10 79 91 89)(11 80 92 90)(12 81 93 85)(31 43 42 53)(32 44 37 54)(33 45 38 49)(34 46 39 50)(35 47 40 51)(36 48 41 52)(55 67 65 77)(56 68 66 78)(57 69 61 73)(58 70 62 74)(59 71 63 75)(60 72 64 76)
(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 91 4 94)(2 96 5 93)(3 95 6 92)(7 15 10 18)(8 14 11 17)(9 13 12 16)(19 80 22 83)(20 79 23 82)(21 84 24 81)(25 89 28 86)(26 88 29 85)(27 87 30 90)(31 78 34 75)(32 77 35 74)(33 76 36 73)(37 67 40 70)(38 72 41 69)(39 71 42 68)(43 56 46 59)(44 55 47 58)(45 60 48 57)(49 64 52 61)(50 63 53 66)(51 62 54 65)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4X 6A ··· 6G 12A ··· 12H order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 D6 C4×S3 kernel C2×C4×Dic3 C4×Dic3 C22×Dic3 C22×C12 C2×Dic3 C2×C12 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 1 4 2 1 8

Matrix representation of C2×C4×Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12 1 0 0 0 12 0
,
 12 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 10 10 0 0 0 7 3

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,1,0],[12,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,10,7,0,0,0,10,3] >;

C2×C4×Dic3 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_3
% in TeX

G:=Group("C2xC4xDic3");
// GroupNames label

G:=SmallGroup(96,129);
// by ID

G=gap.SmallGroup(96,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,86,2309]);
// Polycyclic

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

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