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G = C23⋊C16  order 128 = 27

The semidirect product of C23 and C16 acting via C16/C4=C4

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

Aliases: C23⋊C16, C24.1C8, C22.2M5(2), C22⋊C16⋊1C2, (C2×C8).290D4, (C23×C4).2C4, (C22×C8).5C4, (C22×C4).1C8, C2.2(C23⋊C8), C22.2(C2×C16), C23.26(C2×C8), C4.35(C23⋊C4), C2.3(C22⋊C16), C2.1(C23.C8), (C2×C4).54M4(2), (C22×C8).1C22, C4.20(C4.D4), C22.33(C22⋊C8), (C2×C22⋊C8).7C2, (C22×C4).427(C2×C4), (C2×C4).376(C22⋊C4), SmallGroup(128,46)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23⋊C16
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C22⋊C8 — C23⋊C16
Lower central
C1 — C2 — C22 — C23⋊C16
Upper central
C1 — C2×C4 — C22×C8 — C23⋊C16
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C23⋊C16

Generators and relations for C23⋊C16
 G = < a,b,c,d | a2=b2=c2=d16=1, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

Subgroups: 160 in 70 conjugacy classes, 26 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C16, C2×C8, C2×C8, C22×C4, C22×C4, C24, C22⋊C8, C2×C16, C22×C8, C23×C4, C22⋊C16, C2×C22⋊C8, C23⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C16, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C2×C16, M5(2), C23⋊C8, C22⋊C16, C23.C8, C23⋊C16

Smallest permutation representation of C23⋊C16
►On 32 points
Generators in S32
(1 9)(2 17)(3 26)(5 13)(6 21)(7 30)(10 25)(11 18)(14 29)(15 22)(20 28)(24 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A···8H8I8J8K8L16A···16P
order12222222444444448···8888816···16
size11112244111122442···244444···4

44 irreducible representations

dim11111111222444
type++++++
imageC1C2C2C4C4C8C8C16D4M4(2)M5(2)C23⋊C4C4.D4C23.C8
kernelC23⋊C16C22⋊C16C2×C22⋊C8C22×C8C23×C4C22×C4C24C23C2×C8C2×C4C22C4C4C2
# reps121224416224112

Matrix representation of C23⋊C16 ►in GL6(𝔽17)

1600000
010000
001000
0001600
000010
0000016
,
1600000
0160000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
010000
800000
000010
000001
000100
0016000

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,8,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C23⋊C16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes C_{16}
 
% in TeX
 
G:=Group("C2^3:C16");
 
// GroupNames label
 
G:=SmallGroup(128,46);
 
// by ID
 
G=gap.SmallGroup(128,46);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,346,136,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
 
// generators/relations
 

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