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G = D16⋊C22order 128 = 27

4th semidirect product of D16 and C22 acting via C22/C2=C2

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

Aliases: D164C22, C8.15C24, C16.2C23, Q324C22, D8.4C23, C23.21D8, SD323C22, Q16.4C23, M5(2)⋊7C22, C4○D163C2, C4.77(C2×D8), (C2×C4).56D8, C8.58(C2×D4), C4(C16⋊C22), C16⋊C227C2, (C2×C16)⋊4C22, C4(Q32⋊C2), Q32⋊C27C2, (C2×C8).149D4, C4○D86C22, (C2×D8)⋊53C22, (C2×M5(2))⋊5C2, C22.26(C2×D8), C4.21(C22×D4), C2.30(C22×D8), (C2×C8).293C23, (C2×Q16)⋊57C22, (C22×C4).534D4, (C22×C8).296C22, (C2×C4○D8)⋊28C2, (C2×C4).660(C2×D4), SmallGroup(128,2146)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D16⋊C22
C1C2C4C8C2×C8C22×C8C2×C4○D8 — D16⋊C22
C1C2C4C8 — D16⋊C22
C1C4C22×C4C22×C8 — D16⋊C22
C1C2C2C2C2C4C4C8 — D16⋊C22

Generators and relations for D16⋊C22
 G = < a,b,c,d | a16=b2=c2=d2=1, bab=a-1, cac=a9, ad=da, cbc=dbd=a8b, cd=dc >

Subgroups: 404 in 182 conjugacy classes, 90 normal (16 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×2], SD16 [×8], Q16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C2×C16 [×2], M5(2) [×4], D16 [×4], SD32 [×8], Q32 [×4], C22×C8, C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4○D8 [×8], C4○D8 [×4], C2×C4○D4 [×2], C2×M5(2), C4○D16 [×4], C16⋊C22 [×4], Q32⋊C2 [×4], C2×C4○D8 [×2], D16⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D16⋊C22

Smallest permutation representation of D16⋊C22
On 32 points
Generators in S32
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A···16H
order12222222244444444488888816···16
size1122288881122288882222444···4

32 irreducible representations

dim11111122224
type++++++++++
imageC1C2C2C2C2C2D4D4D8D8D16⋊C22
kernelD16⋊C22C2×M5(2)C4○D16C16⋊C22Q32⋊C2C2×C4○D8C2×C8C22×C4C2×C4C23C1
# reps11444231624

Matrix representation of D16⋊C22 in GL4(𝔽17) generated by

0004
0040
12500
5500
,
0040
0004
13000
01300
,
1000
0100
00160
00016
,
0400
13000
00013
0040
G:=sub<GL(4,GF(17))| [0,0,12,5,0,0,5,5,0,4,0,0,4,0,0,0],[0,0,13,0,0,0,0,13,4,0,0,0,0,4,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0] >;

D16⋊C22 in GAP, Magma, Sage, TeX

D_{16}\rtimes C_2^2
% in TeX

G:=Group("D16:C2^2");
// GroupNames label

G:=SmallGroup(128,2146);
// by ID

G=gap.SmallGroup(128,2146);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1430,248,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^9,a*d=d*a,c*b*c=d*b*d=a^8*b,c*d=d*c>;
// generators/relations

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