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G = C16○D8  order 128 = 27

Central product of C16 and D8

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C16○D8, C16○Q16, C16○SD16, D8.2C8, C16.14D4, Q16.2C8, SD16.2C8, M5(2).24C22, C16○C4≀C2, C4≀C2.2C4, (C4×C16)⋊17C2, C16○(C8○D8), C16○(C4○D8), D4○C16⋊6C2, C8.11(C2×C8), D4.C8⋊7C2, C8○D8.7C2, C4○D8.7C4, D4.3(C2×C8), C2.18(C8×D4), Q8.3(C2×C8), C16○(D4.C8), C4.178(C4×D4), C8.141(C2×D4), C16○(C8.C8), C16○(C8.C4), C8.C8⋊14C2, C8.C4.7C4, C8.60(C4○D4), C4.15(C22×C8), (C2×C8).607C23, C42.274(C2×C4), (C4×C8).421C22, (C2×C16).68C22, C8○D4.17C22, C22.1(C8○D4), M4(2).23(C2×C4), (C2×C8).181(C2×C4), C4○D4.18(C2×C4), (C2×C4).446(C22×C4), SmallGroup(128,902)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C16○D8
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — D4○C16 — C16○D8
Lower central
C1 — C2 — C4 — C16○D8
Upper central
C1 — C16 — C2×C16 — C16○D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16○D8

Generators and relations for C16○D8
 G = < a,b,c | a16=c2=1, b4=a8, ab=ba, ac=ca, cbc=a8b3 >

Subgroups: 104 in 73 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C16, C16, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C16, C2×C16, M5(2), M5(2), C8○D4, C4○D8, C4×C16, D4.C8, C8.C8, C8○D8, D4○C16, C16○D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8×D4, C16○D8

Smallest permutation representation of C16○D8
►On 32 points
Generators in S32
(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)

(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)

(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 17)(13 18)(14 19)(15 20)(16 21)
 
G:=sub<Sym(32)| (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), (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,17)(13,18)(14,19)(15,20)(16,21)>;
 
G:=Group( (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), (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,17)(13,18)(14,19)(15,20)(16,21) );
 
G=PermutationGroup([[(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)], [(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,17),(13,18),(14,19),(15,20),(16,21)]])
 

56 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I8A8B8C8D8E···8J8K8L8M8N16A···16H16I···16T16U···16AB
order12222444···44488888···8888816···1616···1616···16
size11244112···24411112···244441···12···24···4

56 irreducible representations

dim1111111111112222
type+++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D4C4○D4C8○D4C16○D8
kernelC16○D8C4×C16D4.C8C8.C8C8○D8D4○C16C4≀C2C8.C4C4○D8D8SD16Q16C16C8C22C1
# reps11211242248422416

Matrix representation of C16○D8 ►in GL2(𝔽17) generated by

60
06
,
20
09
,
09
20
G:=sub<GL(2,GF(17))| [6,0,0,6],[2,0,0,9],[0,2,9,0] >;
 

C16○D8 in GAP, Magma, Sage, TeX 

C_{16}\circ D_8
 
% in TeX
 
G:=Group("C16oD8");
 
// GroupNames label
 
G:=SmallGroup(128,902);
 
// by ID
 
G=gap.SmallGroup(128,902);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,2019,1018,248,102,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=c^2=1,b^4=a^8,a*b=b*a,a*c=c*a,c*b*c=a^8*b^3>;
 
// generators/relations
 

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