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G = D16.C4order 128 = 27

1st non-split extension by D16 of C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 5), monomial

Aliases: D16.1C4, C16.22D4, C4.18D16, Q32.1C4, C8.20SD16, C22.1SD32, (C2×C32)⋊4C2, (C2×C4).67D8, C16.12(C2×C4), C4○D16.1C2, (C2×C8).265D4, C8.4Q81C2, C2.8(C2.D16), C8.15(C22⋊C4), (C2×C16).92C22, C4.15(D4⋊C4), SmallGroup(128,149)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — D16.C4
C1C2C4C8C2×C8C2×C16C4○D16 — D16.C4
C1C2C4C8C16 — D16.C4
C1C4C2×C4C2×C8C2×C16 — D16.C4
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — D16.C4

Generators and relations for D16.C4
 G = < a,b,c | a16=b2=1, c4=a8, bab=cac-1=a-1, cbc-1=a13b >

2C2
16C2
8C22
8C4
4Q8
4D4
8C2×C4
8D4
8C8
2Q16
2D8
4C4○D4
4SD16
4M4(2)
2C32
2C8.C4
2SD32
2C4○D8

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

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F16A···16H32A···32P
order1222444488888816···1632···32
size1121611216222216162···22···2

38 irreducible representations

dim1111112222222
type++++++++
imageC1C2C2C2C4C4D4D4SD16D8D16SD32D16.C4
kernelD16.C4C8.4Q8C2×C32C4○D16D16Q32C16C2×C8C8C2×C4C4C22C1
# reps11112211224416

Matrix representation of D16.C4 in GL2(𝔽97) generated by

6993
273
,
6993
2628
,
3986
458
G:=sub<GL(2,GF(97))| [69,2,93,73],[69,26,93,28],[39,4,86,58] >;

D16.C4 in GAP, Magma, Sage, TeX

D_{16}.C_4
% in TeX

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

G:=SmallGroup(128,149);
// by ID

G=gap.SmallGroup(128,149);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,219,268,248,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=1,c^4=a^8,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^13*b>;
// generators/relations

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Subgroup lattice of D16.C4 in TeX

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