direct product, metacyclic, nilpotent (class 3), monomial
Aliases: C32×D8, C24⋊3C6, C4.1C62, D4⋊(C3×C6), C8⋊1(C3×C6), (C3×C24)⋊5C2, (C3×D4)⋊4C6, (C3×C6).41D4, C6.20(C3×D4), C12.23(C2×C6), (D4×C32)⋊7C2, C2.3(D4×C32), (C3×C12).50C22, SmallGroup(144,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 114 in 66 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, D4, C32, C12, C2×C6, D8, C3×C6, C3×C6, C24, C3×D4, C3×C12, C62, C3×D8, C3×C24, D4×C32, C32×D8
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, C62, C3×D8, D4×C32, C32×D8
►On 72 points
Generators in S72
(1 34 61)(2 35 62)(3 36 63)(4 37 64)(5 38 57)(6 39 58)(7 40 59)(8 33 60)(9 28 54)(10 29 55)(11 30 56)(12 31 49)(13 32 50)(14 25 51)(15 26 52)(16 27 53)(17 69 42)(18 70 43)(19 71 44)(20 72 45)(21 65 46)(22 66 47)(23 67 48)(24 68 41)
(1 31 17)(2 32 18)(3 25 19)(4 26 20)(5 27 21)(6 28 22)(7 29 23)(8 30 24)(9 47 58)(10 48 59)(11 41 60)(12 42 61)(13 43 62)(14 44 63)(15 45 64)(16 46 57)(33 56 68)(34 49 69)(35 50 70)(36 51 71)(37 52 72)(38 53 65)(39 54 66)(40 55 67)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)
G:=sub<Sym(72)| (1,34,61)(2,35,62)(3,36,63)(4,37,64)(5,38,57)(6,39,58)(7,40,59)(8,33,60)(9,28,54)(10,29,55)(11,30,56)(12,31,49)(13,32,50)(14,25,51)(15,26,52)(16,27,53)(17,69,42)(18,70,43)(19,71,44)(20,72,45)(21,65,46)(22,66,47)(23,67,48)(24,68,41), (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,47,58)(10,48,59)(11,41,60)(12,42,61)(13,43,62)(14,44,63)(15,45,64)(16,46,57)(33,56,68)(34,49,69)(35,50,70)(36,51,71)(37,52,72)(38,53,65)(39,54,66)(40,55,67), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70)>;
G:=Group( (1,34,61)(2,35,62)(3,36,63)(4,37,64)(5,38,57)(6,39,58)(7,40,59)(8,33,60)(9,28,54)(10,29,55)(11,30,56)(12,31,49)(13,32,50)(14,25,51)(15,26,52)(16,27,53)(17,69,42)(18,70,43)(19,71,44)(20,72,45)(21,65,46)(22,66,47)(23,67,48)(24,68,41), (1,31,17)(2,32,18)(3,25,19)(4,26,20)(5,27,21)(6,28,22)(7,29,23)(8,30,24)(9,47,58)(10,48,59)(11,41,60)(12,42,61)(13,43,62)(14,44,63)(15,45,64)(16,46,57)(33,56,68)(34,49,69)(35,50,70)(36,51,71)(37,52,72)(38,53,65)(39,54,66)(40,55,67), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(66,72)(67,71)(68,70) );
G=PermutationGroup([[(1,34,61),(2,35,62),(3,36,63),(4,37,64),(5,38,57),(6,39,58),(7,40,59),(8,33,60),(9,28,54),(10,29,55),(11,30,56),(12,31,49),(13,32,50),(14,25,51),(15,26,52),(16,27,53),(17,69,42),(18,70,43),(19,71,44),(20,72,45),(21,65,46),(22,66,47),(23,67,48),(24,68,41)], [(1,31,17),(2,32,18),(3,25,19),(4,26,20),(5,27,21),(6,28,22),(7,29,23),(8,30,24),(9,47,58),(10,48,59),(11,41,60),(12,42,61),(13,43,62),(14,44,63),(15,45,64),(16,46,57),(33,56,68),(34,49,69),(35,50,70),(36,51,71),(37,52,72),(38,53,65),(39,54,66),(40,55,67)], [(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)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(18,24),(19,23),(20,22),(25,29),(26,28),(30,32),(33,35),(36,40),(37,39),(41,43),(44,48),(45,47),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62),(66,72),(67,71),(68,70)]])
C32×D8 is a maximal subgroup of
C32⋊7D16 C32⋊8SD32 C24⋊8D6 C24.26D6
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4 | 6A | ··· | 6H | 6I | ··· | 6X | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 4 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | C3×D8 |
| kernel | C32×D8 | C3×C24 | D4×C32 | C3×D8 | C24 | C3×D4 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 2 | 8 | 16 |
Matrix representation of C32×D8 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 16 | 57 |
| 0 | 16 | 16 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,64,0,0,0,64],[72,0,0,0,16,16,0,57,16],[72,0,0,0,1,0,0,0,72] >;
C32×D8 in GAP, Magma, Sage, TeX
C_3^2\times D_8
% in TeX
G:=Group("C3^2xD8"); // GroupNames label
G:=SmallGroup(144,106);
// by ID
G=gap.SmallGroup(144,106);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,457,3244,1630,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations