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## G = C2×D6⋊3D4order 192 = 26·3

### Direct product of C2 and D6⋊3D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×D6⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×D6⋊3D4
 Lower central C3 — C2×C6 — C2×D6⋊3D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×D63D4
G = < a,b,c,d,e | a2=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 1096 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C4⋊Dic3, C6.D4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, C2×C4⋊D4, C2×C4⋊Dic3, D63D4, C2×C6.D4, S3×C22×C4, C22×C3⋊D4, D4×C2×C6, C2×D63D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, S3×D4, D42S3, C2×C3⋊D4, S3×C23, C2×C4⋊D4, D63D4, C2×S3×D4, C2×D42S3, C22×C3⋊D4, C2×D63D4

Smallest permutation representation of C2×D63D4
On 96 points
Generators in S96
(1 64)(2 65)(3 66)(4 61)(5 62)(6 63)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 75)(20 76)(21 77)(22 78)(23 73)(24 74)(25 71)(26 72)(27 67)(28 68)(29 69)(30 70)(31 87)(32 88)(33 89)(34 90)(35 85)(36 86)(37 83)(38 84)(39 79)(40 80)(41 81)(42 82)(49 95)(50 96)(51 91)(52 92)(53 93)(54 94)
(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 63)(2 62)(3 61)(4 66)(5 65)(6 64)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 58)(14 57)(15 56)(16 55)(17 60)(18 59)(19 73)(20 78)(21 77)(22 76)(23 75)(24 74)(25 67)(26 72)(27 71)(28 70)(29 69)(30 68)(31 88)(32 87)(33 86)(34 85)(35 90)(36 89)(37 82)(38 81)(39 80)(40 79)(41 84)(42 83)(49 91)(50 96)(51 95)(52 94)(53 93)(54 92)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 75 91 67)(8 76 92 68)(9 77 93 69)(10 78 94 70)(11 73 95 71)(12 74 96 72)(19 51 27 43)(20 52 28 44)(21 53 29 45)(22 54 30 46)(23 49 25 47)(24 50 26 48)(55 87 63 79)(56 88 64 80)(57 89 65 81)(58 90 66 82)(59 85 61 83)(60 86 62 84)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 63)(8 64)(9 65)(10 66)(11 61)(12 62)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 6 6 6 6 2 2 2 2 2 6 6 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 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C2×D6⋊3D4 C2×C4⋊Dic3 D6⋊3D4 C2×C6.D4 S3×C22×C4 C22×C3⋊D4 D4×C2×C6 C22×D4 C2×C12 C22×S3 C22×C4 C2×D4 C24 C2×C6 C2×C4 C22 C22 # reps 1 1 8 2 1 2 1 1 4 4 1 4 2 4 8 2 2

Matrix representation of C2×D63D4 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 1 0
,
 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 12 1
,
 12 0 0 0 0 0 8 0 0 0 0 0 5 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 5 0 0 0 8 0 0 0 0 0 0 11 4 0 0 0 9 2

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,12,0],[1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,1],[12,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,0,8,0,0,0,5,0,0,0,0,0,0,11,9,0,0,0,4,2] >;

C2×D63D4 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes_3D_4
% in TeX

G:=Group("C2xD6:3D4");
// GroupNames label

G:=SmallGroup(192,1359);
// by ID

G=gap.SmallGroup(192,1359);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,6278]);
// Polycyclic

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

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