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## G = C3×C8⋊5D4order 192 = 26·3

### Direct product of C3 and C8⋊5D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8⋊5D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×D4 — C6×SD16 — C3×C8⋊5D4
 Lower central C1 — C2 — C2×C4 — C3×C8⋊5D4
 Upper central C1 — C2×C6 — C4×C12 — C3×C8⋊5D4

Generators and relations for C3×C85D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >

Subgroups: 290 in 142 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×C8, C41D4, C4⋊Q8, C2×SD16, C4×C12, C3×C4⋊C4, C2×C24, C3×SD16, C6×D4, C6×D4, C6×Q8, C85D4, C4×C24, C3×C41D4, C3×C4⋊Q8, C6×SD16, C3×C85D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C3×D4, C22×C6, C41D4, C2×SD16, C3×SD16, C6×D4, C85D4, C3×C41D4, C6×SD16, C3×C85D4

Smallest permutation representation of C3×C85D4
On 96 points
Generators in S96
(1 54 30)(2 55 31)(3 56 32)(4 49 25)(5 50 26)(6 51 27)(7 52 28)(8 53 29)(9 67 83)(10 68 84)(11 69 85)(12 70 86)(13 71 87)(14 72 88)(15 65 81)(16 66 82)(17 73 91)(18 74 92)(19 75 93)(20 76 94)(21 77 95)(22 78 96)(23 79 89)(24 80 90)(33 42 61)(34 43 62)(35 44 63)(36 45 64)(37 46 57)(38 47 58)(39 48 59)(40 41 60)
(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 35 21 71)(2 36 22 72)(3 37 23 65)(4 38 24 66)(5 39 17 67)(6 40 18 68)(7 33 19 69)(8 34 20 70)(9 26 59 91)(10 27 60 92)(11 28 61 93)(12 29 62 94)(13 30 63 95)(14 31 64 96)(15 32 57 89)(16 25 58 90)(41 74 84 51)(42 75 85 52)(43 76 86 53)(44 77 87 54)(45 78 88 55)(46 79 81 56)(47 80 82 49)(48 73 83 50)
(1 71)(2 66)(3 69)(4 72)(5 67)(6 70)(7 65)(8 68)(9 26)(10 29)(11 32)(12 27)(13 30)(14 25)(15 28)(16 31)(17 39)(18 34)(19 37)(20 40)(21 35)(22 38)(23 33)(24 36)(41 76)(42 79)(43 74)(44 77)(45 80)(46 75)(47 78)(48 73)(49 88)(50 83)(51 86)(52 81)(53 84)(54 87)(55 82)(56 85)(57 93)(58 96)(59 91)(60 94)(61 89)(62 92)(63 95)(64 90)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 SD16 C3×D4 C3×D4 C3×SD16 kernel C3×C8⋊5D4 C4×C24 C3×C4⋊1D4 C3×C4⋊Q8 C6×SD16 C8⋊5D4 C4×C8 C4⋊1D4 C4⋊Q8 C2×SD16 C24 C2×C12 C12 C8 C2×C4 C4 # reps 1 1 1 1 4 2 2 2 2 8 4 2 8 8 4 16

Matrix representation of C3×C85D4 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 61 67 0 0 12 0 0 0 0 0 72 2 0 0 72 1
,
 1 0 0 0 0 1 0 0 0 0 1 71 0 0 1 72
,
 1 1 0 0 0 72 0 0 0 0 1 71 0 0 0 72
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[61,12,0,0,67,0,0,0,0,0,72,72,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,1,72,0,0,0,0,1,0,0,0,71,72] >;

C3×C85D4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_5D_4
% in TeX

G:=Group("C3xC8:5D4");
// GroupNames label

G:=SmallGroup(192,925);
// by ID

G=gap.SmallGroup(192,925);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,268,4204,172]);
// Polycyclic

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

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