Copied to
clipboard

G = C2xC16:C4  order 128 = 27

Direct product of C2 and C16:C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2xC16:C4, C8.19C42, M5(2):11C4, C23.33M4(2), M5(2).17C22, (C2xC16):7C4, C16:3(C2xC4), C4o(C16:C4), C8:C4.16C4, C4.5(C8:C4), C42.22(C2xC4), (C2xC42).22C4, (C2xC4).91C42, C8.61(C22xC4), (C22xC8).18C4, C4.43(C2xC42), (C2xC8).382C23, (C2xC4).79M4(2), C4.47(C2xM4(2)), (C2xM5(2)).19C2, C8:C4.148C22, C22.15(C8:C4), (C22xC8).412C22, C22.21(C2xM4(2)), C2.11(C2xC8:C4), (C2xC8).247(C2xC4), (C2xC8:C4).37C2, (C22xC4).485(C2xC4), (C2xC4).556(C22xC4), SmallGroup(128,841)

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

Derived series
C1 — C4 — C2xC16:C4
Chief series
C1 — C2 — C4 — C2xC4 — C2xC8 — C22xC8 — C2xC8:C4 — C2xC16:C4
Lower central
C1 — C4 — C2xC16:C4
Upper central
C1 — C2xC4 — C2xC16:C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2xC8 — C2xC16:C4

Generators and relations for C2xC16:C4
 G = < a,b,c | a2=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >

Subgroups: 108 in 84 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C23, C16, C42, C42, C2xC8, C2xC8, C2xC8, C22xC4, C22xC4, C8:C4, C2xC16, M5(2), C2xC42, C22xC8, C16:C4, C2xC8:C4, C2xM5(2), C2xC16:C4
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, M4(2), C22xC4, C8:C4, C2xC42, C2xM4(2), C16:C4, C2xC8:C4, C2xC16:C4

Smallest permutation representation of C2xC16:C4
►On 32 points
Generators in S32
(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 23)(2 28 10 20)(3 17)(4 22 12 30)(5 27)(6 32 14 24)(7 21)(8 26 16 18)(9 31)(11 25)(13 19)(15 29)
 
G:=sub<Sym(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), (1,23)(2,28,10,20)(3,17)(4,22,12,30)(5,27)(6,32,14,24)(7,21)(8,26,16,18)(9,31)(11,25)(13,19)(15,29)>;
 
G:=Group( (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), (1,23)(2,28,10,20)(3,17)(4,22,12,30)(5,27)(6,32,14,24)(7,21)(8,26,16,18)(9,31)(11,25)(13,19)(15,29) );
 
G=PermutationGroup([[(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)], [(1,23),(2,28,10,20),(3,17),(4,22,12,30),(5,27),(6,32,14,24),(7,21),(8,26,16,18),(9,31),(11,25),(13,19),(15,29)]])
 

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11112211112244442···244444···4

44 irreducible representations

dim111111111224
type++++
imageC1C2C2C2C4C4C4C4C4M4(2)M4(2)C16:C4
kernelC2xC16:C4C16:C4C2xC8:C4C2xM5(2)C8:C4C2xC16M5(2)C2xC42C22xC8C2xC4C23C2
# reps141248822624

Matrix representation of C2xC16:C4 ►in GL6(F17)

1600000
0160000
0016000
0001600
0000160
0000016
,
0160000
400000
000010
0000416
0016900
007100
,
1600000
010000
001000
0041600
0000130
000014

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,9,1,0,0,1,4,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,16,0,0,0,0,0,0,13,1,0,0,0,0,0,4] >;
 

C2xC16:C4 in GAP, Magma, Sage, TeX 

C_2\times C_{16}\rtimes C_4
 
% in TeX
 
G:=Group("C2xC16:C4");
 
// GroupNames label
 
G:=SmallGroup(128,841);
 
// by ID
 
G=gap.SmallGroup(128,841);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,477,120,1018,136,2804,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<