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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×D4 — C6×D8 — C3×C8⋊2D4
 Lower central C1 — C2 — C2×C4 — C3×C8⋊2D4
 Upper central C1 — C2×C6 — C22×C12 — C3×C8⋊2D4

Generators and relations for C3×C82D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 282 in 130 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C3×D8, C22×C12, C6×D4, C6×D4, C82D4, C3×D4⋊C4, C3×C4.Q8, C3×C4⋊D4, C6×M4(2), C6×D8, C3×C82D4
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, C82D4, C3×C4⋊D4, C3×C8⋊C22, C3×C82D4

Smallest permutation representation of C3×C82D4
On 96 points
Generators in S96
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 30 58)(10 31 59)(11 32 60)(12 25 61)(13 26 62)(14 27 63)(15 28 64)(16 29 57)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 54 89)(42 55 90)(43 56 91)(44 49 92)(45 50 93)(46 51 94)(47 52 95)(48 53 96)
(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 25 75 43)(2 28 76 46)(3 31 77 41)(4 26 78 44)(5 29 79 47)(6 32 80 42)(7 27 73 45)(8 30 74 48)(9 34 96 24)(10 37 89 19)(11 40 90 22)(12 35 91 17)(13 38 92 20)(14 33 93 23)(15 36 94 18)(16 39 95 21)(49 68 62 86)(50 71 63 81)(51 66 64 84)(52 69 57 87)(53 72 58 82)(54 67 59 85)(55 70 60 88)(56 65 61 83)
(2 8)(3 7)(4 6)(9 94)(10 93)(11 92)(12 91)(13 90)(14 89)(15 96)(16 95)(18 24)(19 23)(20 22)(25 43)(26 42)(27 41)(28 48)(29 47)(30 46)(31 45)(32 44)(33 37)(34 36)(38 40)(49 60)(50 59)(51 58)(52 57)(53 64)(54 63)(55 62)(56 61)(66 72)(67 71)(68 70)(73 77)(74 76)(78 80)(81 85)(82 84)(86 88)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 6A ··· 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 24A ··· 24H order 1 2 2 2 2 2 2 3 3 4 4 4 4 4 6 ··· 6 6 6 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 4 8 8 1 1 2 2 4 8 8 1 ··· 1 4 4 8 8 8 8 4 4 4 4 2 2 2 2 4 4 8 8 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⋊2D4 C3×D4⋊C4 C3×C4.Q8 C3×C4⋊D4 C6×M4(2) C6×D8 C8⋊2D4 D4⋊C4 C4.Q8 C4⋊D4 C2×M4(2) C2×D8 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×C82D4 in GL8(𝔽73)

 8 0 0 0 0 0 0 0 0 8 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
,
 46 0 0 0 0 0 0 0 71 27 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 0 15 0 0 0 0 59 23 15 58 0 0 0 0 69 4 0 64 0 0 0 0 26 51 23 27
,
 72 27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 71 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 56 17 72 2 0 0 0 0 1 0 0 0 0 0 0 0 72 2 17 56
,
 1 46 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 72 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 1 0 0 0 0 0 17 0 1 72

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[46,71,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,59,69,26,0,0,0,0,50,23,4,51,0,0,0,0,0,15,0,23,0,0,0,0,15,58,64,27],[72,0,0,0,0,0,0,0,27,1,0,0,0,0,0,0,0,0,72,71,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,56,1,72,0,0,0,0,0,17,0,2,0,0,0,0,1,72,0,17,0,0,0,0,0,2,0,56],[1,0,0,0,0,0,0,0,46,72,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,17,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72] >;

C3×C82D4 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_2D_4
% in TeX

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

G:=SmallGroup(192,902);
// by ID

G=gap.SmallGroup(192,902);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,1059,4204,172]);
// 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,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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