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## G = C2×C16⋊C4order 128 = 27

### Direct product of C2 and C16⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C16⋊C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C8⋊C4 — C2×C16⋊C4
 Lower central C1 — C4 — C2×C16⋊C4
 Upper central C1 — C2×C4 — C2×C16⋊C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C16⋊C4

Generators and relations for C2×C16⋊C4
G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >

Subgroups: 108 in 84 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C16, M5(2), C2×C42, C22×C8, C16⋊C4, C2×C8⋊C4, C2×M5(2), C2×C16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C16⋊C4, C2×C8⋊C4, C2×C16⋊C4

Smallest permutation representation of C2×C16⋊C4
On 32 points
Generators in S32
(1 31)(2 32)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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 23)(2 28 10 20)(3 17)(4 22 12 30)(5 27)(6 32 14 24)(7 21)(8 26 16 18)(9 31)(11 25)(13 19)(15 29)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C4 M4(2) M4(2) C16⋊C4 kernel C2×C16⋊C4 C16⋊C4 C2×C8⋊C4 C2×M5(2) C8⋊C4 C2×C16 M5(2) C2×C42 C22×C8 C2×C4 C23 C2 # reps 1 4 1 2 4 8 8 2 2 6 2 4

Matrix representation of C2×C16⋊C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 4 16 0 0 16 9 0 0 0 0 7 1 0 0
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 4 16 0 0 0 0 0 0 13 0 0 0 0 0 1 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,9,1,0,0,1,4,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,16,0,0,0,0,0,0,13,1,0,0,0,0,0,4] >;

C2×C16⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{16}\rtimes C_4
% in TeX

G:=Group("C2xC16:C4");
// GroupNames label

G:=SmallGroup(128,841);
// by ID

G=gap.SmallGroup(128,841);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,477,120,1018,136,2804,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations

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