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G = C4.16D16order 128 = 27

1st central extension by C4 of D16

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

Aliases: D81C8, C4.16D16, C4.14SD32, C8.6M4(2), C42.309D4, C4.1C4≀C2, (C4×C16)⋊2C2, C8.7(C2×C8), C81C81C2, (C2×D8).5C4, (C4×D8).1C2, C2.D8.6C4, C2.6(D4⋊C8), (C2×C4).158D8, (C2×C8).295D4, C4.1(C22⋊C8), (C2×C4).59SD16, C2.1(C2.D16), (C4×C8).384C22, C2.1(D8.C4), C22.40(D4⋊C4), (C2×C8).165(C2×C4), (C2×C4).211(C22⋊C4), SmallGroup(128,63)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4.16D16
C1C2C22C2×C4C42C4×C8C4×D8 — C4.16D16
C1C2C4C8 — C4.16D16
C1C2×C4C42C4×C8 — C4.16D16
C1C2C2C2C2C2×C4C2×C4C4×C8 — C4.16D16

Generators and relations for C4.16D16
 G = < a,b,c | a4=b16=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

8C2
8C2
2C4
4C22
4C22
8C22
8C22
8C4
2C8
2D4
2D4
4D4
4C23
4C2×C4
8C2×C4
8C8
8C2×C4
2C16
2D8
2C2×D4
2C4⋊C4
2C16
4C22⋊C4
4C2×C8
4C22×C4
2C2×C16
2D4⋊C4
2C4×D4
2C4⋊C8

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11118811112222882···288882···2

44 irreducible representations

dim1111111222222222
type++++++++
imageC1C2C2C2C4C4C8D4D4M4(2)D8SD16C4≀C2D16SD32D8.C4
kernelC4.16D16C81C8C4×C16C4×D8C2.D8C2×D8D8C42C2×C8C8C2×C4C2×C4C4C4C4C2
# reps1111228112224448

Matrix representation of C4.16D16 in GL4(𝔽17) generated by

13000
01300
0040
0004
,
10700
101000
0035
00713
,
71000
101000
0048
00713
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[10,10,0,0,7,10,0,0,0,0,3,7,0,0,5,13],[7,10,0,0,10,10,0,0,0,0,4,7,0,0,8,13] >;

C4.16D16 in GAP, Magma, Sage, TeX

C_4._{16}D_{16}
% in TeX

G:=Group("C4.16D16");
// GroupNames label

G:=SmallGroup(128,63);
// by ID

G=gap.SmallGroup(128,63);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,436,136,2804,1411,172]);
// Polycyclic

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

Export

Subgroup lattice of C4.16D16 in TeX

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