p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4○C32, Q8○C32, D4.2C16, C32○M6(2), C32○M5(2), M4(2)○C32, Q8.2C16, M6(2)⋊7C2, C32.7C22, C16.17C23, M4(2).6C8, M5(2).4C4, C4○D4○C32, (C2×C32)⋊9C2, C32○(C8○D4), C4.5(C2×C16), C8.13(C2×C8), C8○D4.6C4, C4○D4.5C8, C32○(D4○C16), C16.15(C2×C4), D4○C16.3C2, C4.38(C22×C8), C2.7(C22×C16), C8.69(C22×C4), C22.1(C2×C16), (C2×C16).107C22, (C2×C4).55(C2×C8), (C2×C8).197(C2×C4), SmallGroup(128,990)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4○C32
G = < a,b,c | a4=b2=1, c16=a2, bab=a-1, ac=ca, bc=cb >
Smallest permutation representation of D4○C32
►On 64 points
Generators in S64
(1 56 17 40)(2 57 18 41)(3 58 19 42)(4 59 20 43)(5 60 21 44)(6 61 22 45)(7 62 23 46)(8 63 24 47)(9 64 25 48)(10 33 26 49)(11 34 27 50)(12 35 28 51)(13 36 29 52)(14 37 30 53)(15 38 31 54)(16 39 32 55)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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)
G:=sub<Sym(64)| (1,56,17,40)(2,57,18,41)(3,58,19,42)(4,59,20,43)(5,60,21,44)(6,61,22,45)(7,62,23,46)(8,63,24,47)(9,64,25,48)(10,33,26,49)(11,34,27,50)(12,35,28,51)(13,36,29,52)(14,37,30,53)(15,38,31,54)(16,39,32,55), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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)>;
G:=Group( (1,56,17,40)(2,57,18,41)(3,58,19,42)(4,59,20,43)(5,60,21,44)(6,61,22,45)(7,62,23,46)(8,63,24,47)(9,64,25,48)(10,33,26,49)(11,34,27,50)(12,35,28,51)(13,36,29,52)(14,37,30,53)(15,38,31,54)(16,39,32,55), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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) );
G=PermutationGroup([[(1,56,17,40),(2,57,18,41),(3,58,19,42),(4,59,20,43),(5,60,21,44),(6,61,22,45),(7,62,23,46),(8,63,24,47),(9,64,25,48),(10,33,26,49),(11,34,27,50),(12,35,28,51),(13,36,29,52),(14,37,30,53),(15,38,31,54),(16,39,32,55)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 16A | ··· | 16H | 16I | ··· | 16T | 32A | ··· | 32P | 32Q | ··· | 32AN |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 | 32 | ··· | 32 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D4○C32 |
| kernel | D4○C32 | C2×C32 | M6(2) | D4○C16 | M5(2) | C8○D4 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 12 | 4 | 24 | 8 | 16 |
Matrix representation of D4○C32 ►in GL2(𝔽97) generated by
| 44 | 50 |
| 2 | 53 |
| 53 | 46 |
| 95 | 44 |
| 78 | 0 |
| 0 | 78 |
G:=sub<GL(2,GF(97))| [44,2,50,53],[53,95,46,44],[78,0,0,78] >;
D4○C32 in GAP, Magma, Sage, TeX
D_4\circ C_{32} % in TeX
G:=Group("D4oC32"); // GroupNames label
G:=SmallGroup(128,990);
// by ID
G=gap.SmallGroup(128,990);
# by ID
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,56,723,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^2=1,c^16=a^2,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations
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