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## G = C16○D8order 128 = 27

### Central product of C16 and D8

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

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 2A 2B 2C 2D 4A 4B 4C ··· 4G 4H 4I 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 16A ··· 16H 16I ··· 16T 16U ··· 16AB order 1 2 2 2 2 4 4 4 ··· 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 16 ··· 16 16 ··· 16 16 ··· 16 size 1 1 2 4 4 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4 C4○D4 C8○D4 C16○D8 kernel C16○D8 C4×C16 D4.C8 C8.C8 C8○D8 D4○C16 C4≀C2 C8.C4 C4○D8 D8 SD16 Q16 C16 C8 C22 C1 # reps 1 1 2 1 1 2 4 2 2 4 8 4 2 2 4 16

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

 6 0 0 6
,
 2 0 0 9
,
 0 9 2 0
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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