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## G = C3×C8.2D4order 192 = 26·3

### Direct product of C3 and C8.2D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8.2D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C6×Q8 — C6×SD16 — C3×C8.2D4
 Lower central C1 — C2 — C2×C4 — C3×C8.2D4
 Upper central C1 — C2×C6 — C4×C12 — C3×C8.2D4

Generators and relations for C3×C8.2D4
G = < a,b,c,d | a3=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >

Subgroups: 226 in 124 conjugacy classes, 58 normal (22 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, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×SD16, C3×Q16, C6×D4, C6×Q8, C6×Q8, C8.2D4, C3×C8⋊C4, C3×C4.4D4, C3×C4⋊Q8, C6×SD16, C6×Q16, C3×C8.2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C41D4, C8.C22, C6×D4, C8.2D4, C3×C41D4, C3×C8.C22, C3×C8.2D4

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

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

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

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

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 12H 12I ··· 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 12 ··· 12 24 ··· 24 size 1 1 1 1 8 1 1 2 2 4 4 8 8 8 1 ··· 1 8 8 4 4 4 4 2 2 2 2 4 4 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 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 C8.C22 C3×C8.C22 kernel C3×C8.2D4 C3×C8⋊C4 C3×C4.4D4 C3×C4⋊Q8 C6×SD16 C6×Q16 C8.2D4 C8⋊C4 C4.4D4 C4⋊Q8 C2×SD16 C2×Q16 C24 C2×C12 C8 C2×C4 C6 C2 # reps 1 1 1 1 2 2 2 2 2 2 4 4 4 2 8 4 2 4

Matrix representation of C3×C8.2D4 in GL8(𝔽73)

 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 71 0 0 0 0 0 0 1 72 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 23 23 50 23 0 0 0 0 50 23 50 50 0 0 0 0 23 50 50 50 0 0 0 0 23 23 23 50
,
 72 2 0 0 0 0 0 0 72 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 23 50 50 50 0 0 0 0 23 23 23 50 0 0 0 0 23 23 50 23 0 0 0 0 50 23 50 50
,
 72 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,23,50,23,23,0,0,0,0,23,23,50,23,0,0,0,0,50,50,50,23,0,0,0,0,23,50,50,50],[72,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,23,23,50,0,0,0,0,50,23,23,23,0,0,0,0,50,23,50,50,0,0,0,0,50,50,23,50],[72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3×C8.2D4 in GAP, Magma, Sage, TeX

C_3\times C_8._2D_4
% in TeX

G:=Group("C3xC8.2D4");
// GroupNames label

G:=SmallGroup(192,930);
// by ID

G=gap.SmallGroup(192,930);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094,1059,268,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
// generators/relations

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