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 [×2], C3 [×4], C4, C22 [×2], C6 [×4], C6 [×8], C8, D4 [×2], C32, C12 [×4], C2×C6 [×8], D8, C3×C6, C3×C6 [×2], C24 [×4], C3×D4 [×8], C3×C12, C62 [×2], C3×D8 [×4], C3×C24, D4×C32 [×2], C32×D8
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], D4, C32, C2×C6 [×4], D8, C3×C6 [×3], C3×D4 [×4], C62, C3×D8 [×4], 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 69 42)(10 70 43)(11 71 44)(12 72 45)(13 65 46)(14 66 47)(15 67 48)(16 68 41)(17 31 51)(18 32 52)(19 25 53)(20 26 54)(21 27 55)(22 28 56)(23 29 49)(24 30 50)
(1 31 9)(2 32 10)(3 25 11)(4 26 12)(5 27 13)(6 28 14)(7 29 15)(8 30 16)(17 42 61)(18 43 62)(19 44 63)(20 45 64)(21 46 57)(22 47 58)(23 48 59)(24 41 60)(33 50 68)(34 51 69)(35 52 70)(36 53 71)(37 54 72)(38 55 65)(39 56 66)(40 49 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)(10 16)(11 15)(12 14)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)(49 53)(50 52)(54 56)(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)(9,69,42)(10,70,43)(11,71,44)(12,72,45)(13,65,46)(14,66,47)(15,67,48)(16,68,41)(17,31,51)(18,32,52)(19,25,53)(20,26,54)(21,27,55)(22,28,56)(23,29,49)(24,30,50), (1,31,9)(2,32,10)(3,25,11)(4,26,12)(5,27,13)(6,28,14)(7,29,15)(8,30,16)(17,42,61)(18,43,62)(19,44,63)(20,45,64)(21,46,57)(22,47,58)(23,48,59)(24,41,60)(33,50,68)(34,51,69)(35,52,70)(36,53,71)(37,54,72)(38,55,65)(39,56,66)(40,49,67), (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)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(49,53)(50,52)(54,56)(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)(9,69,42)(10,70,43)(11,71,44)(12,72,45)(13,65,46)(14,66,47)(15,67,48)(16,68,41)(17,31,51)(18,32,52)(19,25,53)(20,26,54)(21,27,55)(22,28,56)(23,29,49)(24,30,50), (1,31,9)(2,32,10)(3,25,11)(4,26,12)(5,27,13)(6,28,14)(7,29,15)(8,30,16)(17,42,61)(18,43,62)(19,44,63)(20,45,64)(21,46,57)(22,47,58)(23,48,59)(24,41,60)(33,50,68)(34,51,69)(35,52,70)(36,53,71)(37,54,72)(38,55,65)(39,56,66)(40,49,67), (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)(10,16)(11,15)(12,14)(18,24)(19,23)(20,22)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,43)(44,48)(45,47)(49,53)(50,52)(54,56)(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),(9,69,42),(10,70,43),(11,71,44),(12,72,45),(13,65,46),(14,66,47),(15,67,48),(16,68,41),(17,31,51),(18,32,52),(19,25,53),(20,26,54),(21,27,55),(22,28,56),(23,29,49),(24,30,50)], [(1,31,9),(2,32,10),(3,25,11),(4,26,12),(5,27,13),(6,28,14),(7,29,15),(8,30,16),(17,42,61),(18,43,62),(19,44,63),(20,45,64),(21,46,57),(22,47,58),(23,48,59),(24,41,60),(33,50,68),(34,51,69),(35,52,70),(36,53,71),(37,54,72),(38,55,65),(39,56,66),(40,49,67)], [(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),(10,16),(11,15),(12,14),(18,24),(19,23),(20,22),(25,29),(26,28),(30,32),(33,35),(36,40),(37,39),(41,43),(44,48),(45,47),(49,53),(50,52),(54,56),(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