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## G = D4○C32order 128 = 27

### Central product of D4 and C32

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4○C32
 Chief series C1 — C2 — C4 — C8 — C16 — C2×C16 — D4○C16 — D4○C32
 Lower central C1 — C2 — D4○C32
 Upper central C1 — C32 — D4○C32
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — D4○C32

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 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 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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