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G = C3×C8.18D4order 192 = 26·3

Direct product of C3 and C8.18D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8.18D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×Q8 — C6×Q16 — C3×C8.18D4
 Lower central C1 — C2 — C2×C4 — C3×C8.18D4
 Upper central C1 — C2×C6 — C22×C12 — C3×C8.18D4

Generators and relations for C3×C8.18D4
G = < a,b,c,d | a3=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 186 in 114 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C24, C24, C2×C12, C2×C12, C3×Q8, C22×C6, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×Q16, C22×C12, C6×Q8, C8.18D4, C3×Q8⋊C4, C3×C2.D8, C3×C22⋊Q8, C22×C24, C6×Q16, C3×C8.18D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×Q16, C4○D8, C3×Q16, C6×D4, C3×C4○D4, C8.18D4, C3×C4⋊D4, C6×Q16, C3×C4○D8, C3×C8.18D4

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D4 C4○D4 Q16 C3×D4 C3×D4 C3×D4 C4○D8 C3×C4○D4 C3×Q16 C3×C4○D8 kernel C3×C8.18D4 C3×Q8⋊C4 C3×C2.D8 C3×C22⋊Q8 C22×C24 C6×Q16 C8.18D4 Q8⋊C4 C2.D8 C22⋊Q8 C22×C8 C2×Q16 C24 C2×C12 C22×C6 C12 C2×C6 C8 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 2 1 1 2 4 4 2 2 4 4 8 8

Matrix representation of C3×C8.18D4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 63 0 0 0 0 51
,
 0 1 0 0 72 0 0 0 0 0 0 72 0 0 72 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,63,0,0,0,0,51],[0,72,0,0,1,0,0,0,0,0,0,72,0,0,72,0],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,1,0] >;

C3×C8.18D4 in GAP, Magma, Sage, TeX

C_3\times C_8._{18}D_4
% in TeX

G:=Group("C3xC8.18D4");
// GroupNames label

G:=SmallGroup(192,900);
// by ID

G=gap.SmallGroup(192,900);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,4204,172]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations

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