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## G = C3×C8.D4order 192 = 26·3

### Direct product of C3 and C8.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8.D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×Q8 — C6×Q16 — C3×C8.D4
 Lower central C1 — C2 — C2×C4 — C3×C8.D4
 Upper central C1 — C2×C6 — C22×C12 — C3×C8.D4

Generators and relations for C3×C8.D4
G = < a,b,c,d | a3=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 186 in 110 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C24, C24, C2×C12, C2×C12, C3×Q8, C22×C6, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×Q16, C22×C12, C6×Q8, C8.D4, C3×Q8⋊C4, C3×C4.Q8, C3×C22⋊Q8, C6×M4(2), C6×Q16, C3×C8.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C8.C22, C6×D4, C3×C4○D4, C8.D4, C3×C4⋊D4, C3×C8.C22, C3×C8.D4

Smallest permutation representation of C3×C8.D4
On 96 points
Generators in S96
(1 16 33)(2 9 34)(3 10 35)(4 11 36)(5 12 37)(6 13 38)(7 14 39)(8 15 40)(17 53 41)(18 54 42)(19 55 43)(20 56 44)(21 49 45)(22 50 46)(23 51 47)(24 52 48)(25 92 72)(26 93 65)(27 94 66)(28 95 67)(29 96 68)(30 89 69)(31 90 70)(32 91 71)(57 78 85)(58 79 86)(59 80 87)(60 73 88)(61 74 81)(62 75 82)(63 76 83)(64 77 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 69 23 63)(2 72 24 58)(3 67 17 61)(4 70 18 64)(5 65 19 59)(6 68 20 62)(7 71 21 57)(8 66 22 60)(9 25 52 79)(10 28 53 74)(11 31 54 77)(12 26 55 80)(13 29 56 75)(14 32 49 78)(15 27 50 73)(16 30 51 76)(33 89 47 83)(34 92 48 86)(35 95 41 81)(36 90 42 84)(37 93 43 87)(38 96 44 82)(39 91 45 85)(40 94 46 88)
(1 59 5 63)(2 58 6 62)(3 57 7 61)(4 64 8 60)(9 79 13 75)(10 78 14 74)(11 77 15 73)(12 76 16 80)(17 71 21 67)(18 70 22 66)(19 69 23 65)(20 68 24 72)(25 56 29 52)(26 55 30 51)(27 54 31 50)(28 53 32 49)(33 87 37 83)(34 86 38 82)(35 85 39 81)(36 84 40 88)(41 91 45 95)(42 90 46 94)(43 89 47 93)(44 96 48 92)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12N 24A ··· 24H order 1 2 2 2 2 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 1 1 2 2 4 8 8 8 8 1 ··· 1 4 4 4 4 4 4 2 2 2 2 4 4 8 ··· 8 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 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 C3 C6 C6 C6 C6 C6 D4 D4 D4 C4○D4 C3×D4 C3×D4 C3×D4 C3×C4○D4 C8.C22 C3×C8.C22 kernel C3×C8.D4 C3×Q8⋊C4 C3×C4.Q8 C3×C22⋊Q8 C6×M4(2) C6×Q16 C8.D4 Q8⋊C4 C4.Q8 C22⋊Q8 C2×M4(2) C2×Q16 C24 C2×C12 C22×C6 C12 C8 C2×C4 C23 C4 C6 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 2 1 1 2 4 2 2 4 2 4

Matrix representation of C3×C8.D4 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0
,
 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 0 7 68 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 12 72 0 0 0 0 0 0 0 0 12 72 0 0 0 0 0 0 72 61
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 68 66 0 0 0 0 0 0 66 5 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 12 72 0 0 0 0 0 0 0 0 61 1 0 0 0 0 0 0 1 12

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,7,0,0,0,0,0,0,7,68,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,72,0,0,0,0,0,0,0,0,12,72,0,0,0,0,0,0,72,61],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,68,66,0,0,0,0,0,0,66,5,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,72,0,0,0,0,0,0,0,0,61,1,0,0,0,0,0,0,1,12] >;

C3×C8.D4 in GAP, Magma, Sage, TeX

C_3\times C_8.D_4
% in TeX

G:=Group("C3xC8.D4");
// GroupNames label

G:=SmallGroup(192,903);
// by ID

G=gap.SmallGroup(192,903);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,1059,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations

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