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## G = C4×C6.D4order 192 = 26·3

### Direct product of C4 and C6.D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C6.D4
G = < a,b,c,d | a4=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 504 in 258 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×Dic3, C6.D4, C22×Dic3, C22×C12, C22×C12, C22×C12, C23×C6, C4×C22⋊C4, C6.C42, C2×C4×Dic3, C2×C6.D4, C23×C12, C4×C6.D4
Quotients:

Smallest permutation representation of C4×C6.D4
On 96 points
Generators in S96
(1 54 25 47)(2 49 26 48)(3 50 27 43)(4 51 28 44)(5 52 29 45)(6 53 30 46)(7 36 23 40)(8 31 24 41)(9 32 19 42)(10 33 20 37)(11 34 21 38)(12 35 22 39)(13 74 91 67)(14 75 92 68)(15 76 93 69)(16 77 94 70)(17 78 95 71)(18 73 96 72)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(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 8 67)(2 66 9 72)(3 65 10 71)(4 64 11 70)(5 63 12 69)(6 62 7 68)(13 54 80 31)(14 53 81 36)(15 52 82 35)(16 51 83 34)(17 50 84 33)(18 49 79 32)(19 73 26 55)(20 78 27 60)(21 77 28 59)(22 76 29 58)(23 75 30 57)(24 74 25 56)(37 95 43 89)(38 94 44 88)(39 93 45 87)(40 92 46 86)(41 91 47 85)(42 96 48 90)
(1 64 4 61)(2 63 5 66)(3 62 6 65)(7 71 10 68)(8 70 11 67)(9 69 12 72)(13 31 16 34)(14 36 17 33)(15 35 18 32)(19 76 22 73)(20 75 23 78)(21 74 24 77)(25 59 28 56)(26 58 29 55)(27 57 30 60)(37 92 40 95)(38 91 41 94)(39 96 42 93)(43 86 46 89)(44 85 47 88)(45 90 48 87)(49 82 52 79)(50 81 53 84)(51 80 54 83)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 4M ··· 4AB 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 S3 D4 Dic3 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 kernel C4×C6.D4 C6.C42 C2×C4×Dic3 C2×C6.D4 C23×C12 C6.D4 C22×C12 C23×C4 C2×C12 C22×C4 C22×C4 C24 C2×C6 C2×C4 C23 C22 # reps 1 2 2 2 1 16 8 1 4 4 2 1 4 8 8 8

Matrix representation of C4×C6.D4 in GL4(𝔽13) generated by

 1 0 0 0 0 8 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 1 0 0 0 0 10 0 0 0 0 4
,
 8 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
,
 8 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,1,0,0,0,0,10,0,0,0,0,4],[8,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[8,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0] >;

C4×C6.D4 in GAP, Magma, Sage, TeX

C_4\times C_6.D_4
% in TeX

G:=Group("C4xC6.D4");
// GroupNames label

G:=SmallGroup(192,768);
// by ID

G=gap.SmallGroup(192,768);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,100,6278]);
// Polycyclic

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

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