Copied to
clipboard

G = C2×C16⋊C4  order 128 = 27

Direct product of C2 and C16⋊C4

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

Aliases: C2×C16⋊C4, C8.19C42, M5(2)⋊11C4, C23.33M4(2), M5(2).17C22, (C2×C16)⋊7C4, C16⋊3(C2×C4), C4○(C16⋊C4), C8⋊C4.16C4, C4.5(C8⋊C4), C42.22(C2×C4), (C2×C42).22C4, (C2×C4).91C42, C8.61(C22×C4), (C22×C8).18C4, C4.43(C2×C42), (C2×C8).382C23, (C2×C4).79M4(2), C4.47(C2×M4(2)), (C2×M5(2)).19C2, C8⋊C4.148C22, C22.15(C8⋊C4), (C22×C8).412C22, C22.21(C2×M4(2)), C2.11(C2×C8⋊C4), (C2×C8).247(C2×C4), (C2×C8⋊C4).37C2, (C22×C4).485(C2×C4), (C2×C4).556(C22×C4), SmallGroup(128,841)

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

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

Generators and relations for C2×C16⋊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, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C16, M5(2), C2×C42, C22×C8, C16⋊C4, C2×C8⋊C4, C2×M5(2), C2×C16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C16⋊C4, C2×C8⋊C4, C2×C16⋊C4

Smallest permutation representation of C2×C16⋊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
kernelC2×C16⋊C4C16⋊C4C2×C8⋊C4C2×M5(2)C8⋊C4C2×C16M5(2)C2×C42C22×C8C2×C4C23C2
# reps141248822624

Matrix representation of C2×C16⋊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] >;
 

C2×C16⋊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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁