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## G = C4×D16order 128 = 27

### Direct product of C4 and D16

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C4×D16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×D8 — C4×D16
 Lower central C1 — C2 — C4 — C8 — C4×D16
 Upper central C1 — C2×C4 — C42 — C4×C8 — C4×D16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C4×D16

Generators and relations for C4×D16
G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 244 in 87 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C2×C16, D16, C4×D4, C2×D8, C4×C16, C2.D16, C163C4, C4×D8, C2×D16, C4×D16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, D16, C4×D4, C2×D8, C4○D8, C4×D8, C2×D16, C4○D16, C4×D16

Smallest permutation representation of C4×D16
On 64 points
Generators in S64
(1 35 23 49)(2 36 24 50)(3 37 25 51)(4 38 26 52)(5 39 27 53)(6 40 28 54)(7 41 29 55)(8 42 30 56)(9 43 31 57)(10 44 32 58)(11 45 17 59)(12 46 18 60)(13 47 19 61)(14 48 20 62)(15 33 21 63)(16 34 22 64)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(33 51)(34 50)(35 49)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)

G:=sub<Sym(64)| (1,35,23,49)(2,36,24,50)(3,37,25,51)(4,38,26,52)(5,39,27,53)(6,40,28,54)(7,41,29,55)(8,42,30,56)(9,43,31,57)(10,44,32,58)(11,45,17,59)(12,46,18,60)(13,47,19,61)(14,48,20,62)(15,33,21,63)(16,34,22,64), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(33,51)(34,50)(35,49)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)>;

G:=Group( (1,35,23,49)(2,36,24,50)(3,37,25,51)(4,38,26,52)(5,39,27,53)(6,40,28,54)(7,41,29,55)(8,42,30,56)(9,43,31,57)(10,44,32,58)(11,45,17,59)(12,46,18,60)(13,47,19,61)(14,48,20,62)(15,33,21,63)(16,34,22,64), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(33,51)(34,50)(35,49)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52) );

G=PermutationGroup([[(1,35,23,49),(2,36,24,50),(3,37,25,51),(4,38,26,52),(5,39,27,53),(6,40,28,54),(7,41,29,55),(8,42,30,56),(9,43,31,57),(10,44,32,58),(11,45,17,59),(12,46,18,60),(13,47,19,61),(14,48,20,62),(15,33,21,63),(16,34,22,64)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(33,51),(34,50),(35,49),(36,64),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52)]])

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 D8 D16 C4○D8 C4○D16 kernel C4×D16 C4×C16 C2.D16 C16⋊3C4 C4×D8 C2×D16 D16 C42 C2×C8 C8 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 1 2 4 8 4 8

Matrix representation of C4×D16 in GL3(𝔽17) generated by

 13 0 0 0 1 0 0 0 1
,
 16 0 0 0 13 11 0 6 13
,
 1 0 0 0 1 0 0 0 16
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[16,0,0,0,13,6,0,11,13],[1,0,0,0,1,0,0,0,16] >;

C4×D16 in GAP, Magma, Sage, TeX

C_4\times D_{16}
% in TeX

G:=Group("C4xD16");
// GroupNames label

G:=SmallGroup(128,904);
// by ID

G=gap.SmallGroup(128,904);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,1123,570,360,4037,2028,124]);
// Polycyclic

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

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