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## G = C2×D6.D4order 192 = 26·3

### Direct product of C2 and D6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×D6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×D6.D4
 Lower central C3 — C2×C6 — C2×D6.D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×D6.D4
G = < a,b,c,d,e | a2=b6=c2=d4=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 952 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C22×C12, S3×C23, C2×C22.D4, D6.D4, C2×Dic3⋊C4, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, C22×D12, C2×D6.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22.D4, C22×D4, C2×C4○D4, C4○D12, S3×D4, Q83S3, S3×C23, C2×C22.D4, D6.D4, C2×C4○D12, C2×S3×D4, C2×Q83S3, C2×D6.D4

Smallest permutation representation of C2×D6.D4
On 96 points
Generators in S96
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(13 57)(14 58)(15 59)(16 60)(17 55)(18 56)(19 77)(20 78)(21 73)(22 74)(23 75)(24 76)(25 69)(26 70)(27 71)(28 72)(29 67)(30 68)(31 89)(32 90)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(41 79)(42 80)(49 93)(50 94)(51 95)(52 96)(53 91)(54 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 51)(2 50)(3 49)(4 54)(5 53)(6 52)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 44)(14 43)(15 48)(16 47)(17 46)(18 45)(19 41)(20 40)(21 39)(22 38)(23 37)(24 42)(25 35)(26 34)(27 33)(28 32)(29 31)(30 36)(61 92)(62 91)(63 96)(64 95)(65 94)(66 93)(67 89)(68 88)(69 87)(70 86)(71 85)(72 90)(73 83)(74 82)(75 81)(76 80)(77 79)(78 84)
(1 76 16 72)(2 77 17 67)(3 78 18 68)(4 73 13 69)(5 74 14 70)(6 75 15 71)(7 35 95 39)(8 36 96 40)(9 31 91 41)(10 32 92 42)(11 33 93 37)(12 34 94 38)(19 55 29 65)(20 56 30 66)(21 57 25 61)(22 58 26 62)(23 59 27 63)(24 60 28 64)(43 89 53 79)(44 90 54 80)(45 85 49 81)(46 86 50 82)(47 87 51 83)(48 88 52 84)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 96 16 93)(14 91 17 94)(15 92 18 95)(19 82 22 79)(20 83 23 80)(21 84 24 81)(25 88 28 85)(26 89 29 86)(27 90 30 87)(31 67 34 70)(32 68 35 71)(33 69 36 72)(37 73 40 76)(38 74 41 77)(39 75 42 78)(43 65 46 62)(44 66 47 63)(45 61 48 64)(49 57 52 60)(50 58 53 55)(51 59 54 56)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C4○D12 S3×D4 Q8⋊3S3 kernel C2×D6.D4 D6.D4 C2×Dic3⋊C4 C2×D6⋊C4 C6×C4⋊C4 S3×C22×C4 C22×D12 C2×C4⋊C4 C22×S3 C4⋊C4 C22×C4 C2×C6 C22 C22 C22 # reps 1 8 1 3 1 1 1 1 4 4 3 8 8 2 2

Matrix representation of C2×D6.D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 3 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 0 8 0 0 0 0 5 0
,
 8 0 0 0 0 0 12 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 1 3 0 0 0 0 8 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 8 0

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

C2×D6.D4 in GAP, Magma, Sage, TeX

C_2\times D_6.D_4
% in TeX

G:=Group("C2xD6.D4");
// GroupNames label

G:=SmallGroup(192,1064);
// by ID

G=gap.SmallGroup(192,1064);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,675,297,136,6278]);
// Polycyclic

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

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