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G = C4×D16order 128 = 27

Direct product of C4 and D16

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

Aliases: C4×D16, C42.329D4, C167(C2×C4), (C4×C16)⋊8C2, D82(C2×C4), (C4×D8)⋊1C2, C2.13(C4×D8), C2.3(C2×D16), C4.25(C4×D4), C42(C163C4), C163C414C2, (C2×D16).7C2, (C2×C4).172D8, (C2×C8).231D4, C42(C2.D16), C2.D1621C2, C4.11(C4○D8), C2.3(C4○D16), C8.38(C4○D4), C8.35(C22×C4), C22.61(C2×D8), (C4×C8).396C22, (C2×C16).69C22, (C2×C8).502C23, (C2×D8).103C22, C2.D8.150C22, (C2×C4).768(C2×D4), SmallGroup(128,904)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4×D16
C1C2C4C2×C4C2×C8C4×C8C4×D8 — C4×D16
C1C2C4C8 — C4×D16
C1C2×C4C42C4×C8 — C4×D16
C1C2C2C2C2C4C4C2×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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A···8H16A···16P
order122222224444444444448···816···16
size111188881111222288882···22···2

44 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4D4D4C4○D4D8D16C4○D8C4○D16
kernelC4×D16C4×C16C2.D16C163C4C4×D8C2×D16D16C42C2×C8C8C2×C4C4C4C2
# reps11212181124848

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

1300
010
001
,
1600
01311
0613
,
100
010
0016
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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