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G = D4.C16order 128 = 27

The non-split extension by D4 of C16 acting via C16/C8=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D4.C16, Q8.C16, C16.28D4, M6(2)⋊4C2, M4(2).4C8, M5(2).3C4, C8.10M4(2), C22.1M5(2), (C2×C32)⋊2C2, C4.3(C2×C16), C8○D4.2C4, C4○D4.1C8, D4○C16.2C2, C4.35(C22⋊C8), C8.59(C22⋊C4), C2.8(C22⋊C16), (C2×C16).101C22, (C2×C4).48(C2×C8), (C2×C8).189(C2×C4), SmallGroup(128,133)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4.C16
C1C2C4C8C16C2×C16D4○C16 — D4.C16
C1C2C4 — D4.C16
C1C16C2×C16 — D4.C16
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C2×C16 — D4.C16

Generators and relations for D4.C16
 G = < a,b,c | a4=b2=1, c16=a2, bab=a-1, ac=ca, cbc-1=ab >

2C2
4C2
2C22
2C4
2D4
2C2×C4
2C8
2M4(2)
2C2×C8
2C16
2C32
2C32
2C2×C16
2M5(2)

Smallest permutation representation of D4.C16
On 64 points
Generators in S64
(1 54 17 38)(2 55 18 39)(3 56 19 40)(4 57 20 41)(5 58 21 42)(6 59 22 43)(7 60 23 44)(8 61 24 45)(9 62 25 46)(10 63 26 47)(11 64 27 48)(12 33 28 49)(13 34 29 50)(14 35 30 51)(15 36 31 52)(16 37 32 53)
(1 38)(2 18)(3 56)(5 42)(6 22)(7 60)(9 46)(10 26)(11 64)(13 50)(14 30)(15 36)(17 54)(19 40)(21 58)(23 44)(25 62)(27 48)(29 34)(31 52)(33 49)(37 53)(41 57)(45 61)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H16A···16H16I16J16K16L16M16N16O16P32A···32P32Q···32X
order122244448888888816···16161616161616161632···3232···32
size11241124111122441···1222244442···24···4

56 irreducible representations

dim11111111112222
type+++++
imageC1C2C2C2C4C4C8C8C16C16D4M4(2)M5(2)D4.C16
kernelD4.C16C2×C32M6(2)D4○C16M5(2)C8○D4M4(2)C4○D4D4Q8C16C8C22C1
# reps111122448822416

Matrix representation of D4.C16 in GL2(𝔽97) generated by

7943
1518
,
7943
9418
,
4632
4557
G:=sub<GL(2,GF(97))| [79,15,43,18],[79,94,43,18],[46,45,32,57] >;

D4.C16 in GAP, Magma, Sage, TeX

D_4.C_{16}
% in TeX

G:=Group("D4.C16");
// GroupNames label

G:=SmallGroup(128,133);
// by ID

G=gap.SmallGroup(128,133);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,723,352,1018,80,102,124]);
// Polycyclic

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

Export

Subgroup lattice of D4.C16 in TeX

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