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## G = C3×C2.D8order 96 = 25·3

### Direct product of C3 and C2.D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C2.D8
 Lower central C1 — C2 — C4 — C3×C2.D8
 Upper central C1 — C2×C6 — C2×C12 — C3×C2.D8

Generators and relations for C3×C2.D8
G = < a,b,c,d | a3=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of C3×C2.D8
Regular action on 96 points
Generators in S96
(1 36 9)(2 37 10)(3 38 11)(4 39 12)(5 40 13)(6 33 14)(7 34 15)(8 35 16)(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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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 72 19 58)(2 71 20 57)(3 70 21 64)(4 69 22 63)(5 68 23 62)(6 67 24 61)(7 66 17 60)(8 65 18 59)(9 92 43 86)(10 91 44 85)(11 90 45 84)(12 89 46 83)(13 96 47 82)(14 95 48 81)(15 94 41 88)(16 93 42 87)(25 55 79 36)(26 54 80 35)(27 53 73 34)(28 52 74 33)(29 51 75 40)(30 50 76 39)(31 49 77 38)(32 56 78 37)

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

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

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

42 conjugacy classes

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

42 irreducible representations

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

Matrix representation of C3×C2.D8 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 16 57 0 0 0 0 16 16
,
 49 70 0 0 0 0 70 24 0 0 0 0 0 0 32 56 0 0 0 0 56 41 0 0 0 0 0 0 43 62 0 0 0 0 62 30

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[49,70,0,0,0,0,70,24,0,0,0,0,0,0,32,56,0,0,0,0,56,41,0,0,0,0,0,0,43,62,0,0,0,0,62,30] >;

C3×C2.D8 in GAP, Magma, Sage, TeX

C_3\times C_2.D_8
% in TeX

G:=Group("C3xC2.D8");
// GroupNames label

G:=SmallGroup(96,57);
// by ID

G=gap.SmallGroup(96,57);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,367,1443,117]);
// Polycyclic

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

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