Copied to
clipboard

G = D4○C32order 128 = 27

Central product of D4 and C32

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4C32, Q8C32, D4.2C16, C32M6(2), C32M5(2), M4(2)C32, Q8.2C16, M6(2)⋊7C2, C32.7C22, C16.17C23, M4(2).6C8, M5(2).4C4, C4○D4C32, (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

C1C2 — D4○C32
C1C2C4C8C16C2×C16D4○C16 — D4○C32
C1C2 — D4○C32
C1C32 — D4○C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — D4○C32

Generators and relations for D4○C32
 G = < a,b,c | a4=b2=1, c16=a2, bab=a-1, ac=ca, bc=cb >

2C2
2C2
2C2

Smallest permutation representation of D4○C32
On 64 points
Generators in S64
(1 57 17 41)(2 58 18 42)(3 59 19 43)(4 60 20 44)(5 61 21 45)(6 62 22 46)(7 63 23 47)(8 64 24 48)(9 33 25 49)(10 34 26 50)(11 35 27 51)(12 36 28 52)(13 37 29 53)(14 38 30 54)(15 39 31 55)(16 40 32 56)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(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)| (1,57,17,41)(2,58,18,42)(3,59,19,43)(4,60,20,44)(5,61,21,45)(6,62,22,46)(7,63,23,47)(8,64,24,48)(9,33,25,49)(10,34,26,50)(11,35,27,51)(12,36,28,52)(13,37,29,53)(14,38,30,54)(15,39,31,55)(16,40,32,56), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40), (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( (1,57,17,41)(2,58,18,42)(3,59,19,43)(4,60,20,44)(5,61,21,45)(6,62,22,46)(7,63,23,47)(8,64,24,48)(9,33,25,49)(10,34,26,50)(11,35,27,51)(12,36,28,52)(13,37,29,53)(14,38,30,54)(15,39,31,55)(16,40,32,56), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40), (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([(1,57,17,41),(2,58,18,42),(3,59,19,43),(4,60,20,44),(5,61,21,45),(6,62,22,46),(7,63,23,47),(8,64,24,48),(9,33,25,49),(10,34,26,50),(11,35,27,51),(12,36,28,52),(13,37,29,53),(14,38,30,54),(15,39,31,55),(16,40,32,56)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40)], [(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 2A2B2C2D4A4B4C4D4E8A8B8C8D8E···8J16A···16H16I···16T32A···32P32Q···32AN
order122224444488888···816···1616···1632···3232···32
size112221122211112···21···12···21···12···2

80 irreducible representations

dim11111111112
type++++
imageC1C2C2C2C4C4C8C8C16C16D4○C32
kernelD4○C32C2×C32M6(2)D4○C16M5(2)C8○D4M4(2)C4○D4D4Q8C1
# reps13316212424816

Matrix representation of D4○C32 in GL2(𝔽97) generated by

4450
253
,
5346
9544
,
780
078
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

Export

Subgroup lattice of D4○C32 in TeX

׿
×
𝔽