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G = D165C4order 128 = 27

4th semidirect product of D16 and C4 acting via C4/C2=C2

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

Aliases: D165C4, Q325C4, C8.33D8, SD324C4, C42.135D4, C8○D82C2, C16.9(C2×C4), C165C43C2, D8.5(C2×C4), C2.20(C4×D8), C4.32(C4×D4), C4.93(C2×D8), C4○D16.3C2, (C2×C8).213D4, Q16.5(C2×C4), C8.4Q84C2, D8.C46C2, C8.59(C4○D4), C8.42(C22×C4), (C4×C8).161C22, (C2×C8).581C23, (C2×C16).22C22, C4○D8.15C22, C22.2(C4○D8), C8.C4.17C22, (C2×C4).775(C2×D4), SmallGroup(128,911)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D165C4
C1C2C4C2×C4C2×C8C4×C8C8○D8 — D165C4
C1C2C4C8 — D165C4
C1C4C2×C8C4×C8 — D165C4
C1C2C2C2C2C4C4C2×C8 — D165C4

Generators and relations for D165C4
 G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a4b >

Subgroups: 148 in 72 conjugacy classes, 38 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, D4, Q8, C16, C16, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C16, D16, SD32, Q32, C8○D4, C4○D8, C165C4, D8.C4, C8.4Q8, C8○D8, C4○D16, D165C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, D165C4

Smallest permutation representation of D165C4
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 18)(2 17)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
(2 10)(4 12)(6 14)(8 16)(17 29 25 21)(18 22 26 30)(19 31 27 23)(20 24 28 32)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G8A···8H8I8J8K8L16A···16H
order1222244444448···8888816···16
size1128811244882···288884···4

32 irreducible representations

dim111111111222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4D4D4D8C4○D4C4○D8D165C4
kernelD165C4C165C4D8.C4C8.4Q8C8○D8C4○D16D16SD32Q32C42C2×C8C8C8C22C1
# reps112121242114244

Matrix representation of D165C4 in GL4(𝔽17) generated by

00015
00150
01300
1000
,
00150
00015
8000
0800
,
1000
0400
00130
00016
G:=sub<GL(4,GF(17))| [0,0,0,1,0,0,13,0,0,15,0,0,15,0,0,0],[0,0,8,0,0,0,0,8,15,0,0,0,0,15,0,0],[1,0,0,0,0,4,0,0,0,0,13,0,0,0,0,16] >;

D165C4 in GAP, Magma, Sage, TeX

D_{16}\rtimes_5C_4
% in TeX

G:=Group("D16:5C4");
// GroupNames label

G:=SmallGroup(128,911);
// by ID

G=gap.SmallGroup(128,911);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,1123,570,360,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^4*b>;
// generators/relations

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