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## G = C3×D32order 192 = 26·3

### Direct product of C3 and D32

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C3×D32, C963C2, C321C6, D161C6, C24.64D4, C6.15D16, C12.39D8, C48.19C22, C8.5(C3×D4), C4.1(C3×D8), (C3×D16)⋊5C2, C16.2(C2×C6), C2.3(C3×D16), SmallGroup(192,177)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C16 — C3×D32
 Chief series C1 — C2 — C4 — C8 — C16 — C48 — C3×D16 — C3×D32
 Lower central C1 — C2 — C4 — C8 — C16 — C3×D32
 Upper central C1 — C6 — C12 — C24 — C48 — C3×D32

Generators and relations for C3×D32
G = < a,b,c | a3=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C3×D32
On 96 points
Generators in S96
(1 92 62)(2 93 63)(3 94 64)(4 95 33)(5 96 34)(6 65 35)(7 66 36)(8 67 37)(9 68 38)(10 69 39)(11 70 40)(12 71 41)(13 72 42)(14 73 43)(15 74 44)(16 75 45)(17 76 46)(18 77 47)(19 78 48)(20 79 49)(21 80 50)(22 81 51)(23 82 52)(24 83 53)(25 84 54)(26 85 55)(27 86 56)(28 87 57)(29 88 58)(30 89 59)(31 90 60)(32 91 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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(59 64)(60 63)(61 62)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(87 96)(88 95)(89 94)(90 93)(91 92)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 16A 16B 16C 16D 24A 24B 24C 24D 32A ··· 32H 48A ··· 48H 96A ··· 96P order 1 2 2 2 3 3 4 6 6 6 6 6 6 8 8 12 12 16 16 16 16 24 24 24 24 32 ··· 32 48 ··· 48 96 ··· 96 size 1 1 16 16 1 1 2 1 1 16 16 16 16 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 D16 C3×D8 D32 C3×D16 C3×D32 kernel C3×D32 C96 C3×D16 D32 C32 D16 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 8 8 16

Matrix representation of C3×D32 in GL2(𝔽31) generated by

 25 0 0 25
,
 0 6 5 10
,
 1 2 0 30
G:=sub<GL(2,GF(31))| [25,0,0,25],[0,5,6,10],[1,0,2,30] >;

C3×D32 in GAP, Magma, Sage, TeX

C_3\times D_{32}
% in TeX

G:=Group("C3xD32");
// GroupNames label

G:=SmallGroup(192,177);
// by ID

G=gap.SmallGroup(192,177);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,197,1011,514,192,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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