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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

C1C16 — C3×D32
C1C2C4C8C16C48C3×D16 — C3×D32
C1C2C4C8C16 — C3×D32
C1C6C12C24C48 — C3×D32

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

16C2
16C2
8C22
8C22
16C6
16C6
4D4
4D4
8C2×C6
8C2×C6
2D8
2D8
4C3×D4
4C3×D4
2C3×D8
2C3×D8

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 2A2B2C3A3B 4 6A6B6C6D6E6F8A8B12A12B16A16B16C16D24A24B24C24D32A···32H48A···48H96A···96P
order1222334666666881212161616162424242432···3248···4896···96
size11161611211161616162222222222222···22···22···2

57 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4D8C3×D4D16C3×D8D32C3×D16C3×D32
kernelC3×D32C96C3×D16D32C32D16C24C12C8C6C4C3C2C1
# reps112224122448816

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

250
025
,
06
510
,
12
030
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

Export

Subgroup lattice of C3×D32 in TeX

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