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## G = C2×Dic3.D4order 192 = 26·3

### Direct product of C2 and Dic3.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×Dic3.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C2×Dic3.D4
 Lower central C3 — C2×C6 — C2×Dic3.D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×Dic3.D4
G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede=b3d-1 >

Subgroups: 712 in 322 conjugacy classes, 135 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C2×C22⋊Q8, Dic3.D4, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C22×Dic6, C23×Dic3, C2×Dic3.D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, Dic6, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×Dic6, S3×D4, D42S3, S3×C23, C2×C22⋊Q8, Dic3.D4, C22×Dic6, C2×S3×D4, C2×D42S3, C2×Dic3.D4

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 C4○D4 Dic6 S3×D4 D4⋊2S3 kernel C2×Dic3.D4 Dic3.D4 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C22×Dic6 C23×Dic3 C2×C22⋊C4 C2×Dic3 C22×C6 C22⋊C4 C22×C4 C24 C2×C6 C23 C22 C22 # reps 1 8 2 1 1 1 1 1 1 4 4 4 2 1 4 8 2 2

Matrix representation of C2×Dic3.D4 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 7 3 0 0 0 0 0 10 6 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,7,10,0,0,0,0,0,3,6,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12] >;

C2×Dic3.D4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic3.D4");
// GroupNames label

G:=SmallGroup(192,1040);
// by ID

G=gap.SmallGroup(192,1040);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,675,297,80,6278]);
// Polycyclic

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

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