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G = D4○D16order 128 = 27

Central product of D4 and D16

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

Aliases: D4D16, Q8Q32, D4.12D8, Q8.12D8, D165C22, C8.16C24, C16.3C23, Q327C22, D8.5C23, SD324C22, Q16.5C23, M4(2).21D4, M5(2)⋊8C22, D4○D85C2, D4○C163C2, C4○D164C2, C8.15(C2×D4), C4.49(C2×D8), (C2×D16)⋊13C2, C16⋊C225C2, (C2×C16)⋊5C22, C4○D4.35D4, C4○D81C22, C22.6(C2×D8), (C2×D8)⋊33C22, C2.31(C22×D8), C4.22(C22×D4), (C2×C8).294C23, C8○D4.12C22, (C2×C4).184(C2×D4), SmallGroup(128,2147)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D4○D16
C1C2C4C8C2×C8C8○D4D4○D8 — D4○D16
C1C2C4C8 — D4○D16
C1C2C4○D4C8○D4 — D4○D16
C1C2C2C2C2C4C4C8 — D4○D16

Generators and relations for D4○D16
 G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c7 >

Subgroups: 480 in 185 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8 [×6], D8 [×6], SD16 [×6], Q16 [×2], C2×D4 [×12], C4○D4, C4○D4 [×8], C2×C16 [×3], M5(2) [×3], D16 [×9], SD32 [×6], Q32, C8○D4, C2×D8 [×6], C4○D8 [×6], C8⋊C22 [×6], 2+ 1+4 [×2], D4○C16, C2×D16 [×3], C4○D16 [×3], C16⋊C22 [×6], D4○D8 [×2], D4○D16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D4○D16

Smallest permutation representation of D4○D16
On 32 points
Generators in S32
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 31)(2 32)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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)
(1 19)(2 18)(3 17)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)

G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,31)(2,32)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (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), (1,19)(2,18)(3,17)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)>;

G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,31)(2,32)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (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), (1,19)(2,18)(3,17)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20) );

G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,31),(2,32),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30)], [(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)], [(1,19),(2,18),(3,17),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20)])

32 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F8A8B8C8D8E16A16B16C16D16E···16J
order122222···2444444888881616161616···16
size112228···82222882244422224···4

32 irreducible representations

dim11111122224
type+++++++++++
imageC1C2C2C2C2C2D4D4D8D8D4○D16
kernelD4○D16D4○C16C2×D16C4○D16C16⋊C22D4○D8M4(2)C4○D4D4Q8C1
# reps11336231624

Matrix representation of D4○D16 in GL4(𝔽17) generated by

01600
1000
0101
160160
,
016015
1020
0101
160160
,
41100
6400
00411
0064
,
314611
14141111
00143
0033
G:=sub<GL(4,GF(17))| [0,1,0,16,16,0,1,0,0,0,0,16,0,0,1,0],[0,1,0,16,16,0,1,0,0,2,0,16,15,0,1,0],[4,6,0,0,11,4,0,0,0,0,4,6,0,0,11,4],[3,14,0,0,14,14,0,0,6,11,14,3,11,11,3,3] >;

D4○D16 in GAP, Magma, Sage, TeX

D_4\circ D_{16}
% in TeX

G:=Group("D4oD16");
// GroupNames label

G:=SmallGroup(128,2147);
// by ID

G=gap.SmallGroup(128,2147);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,521,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^8=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^7>;
// generators/relations

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