Copied to
clipboard

G = C2×C16⋊C4order 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, C163(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

C1C4 — C2×C16⋊C4
C1C2C4C2×C4C2×C8C22×C8C2×C8⋊C4 — C2×C16⋊C4
C1C4 — C2×C16⋊C4
C1C2×C4 — C2×C16⋊C4
C1C2C2C2C2C4C4C2×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 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C16 [×8], C42 [×2], C42, C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4, C8⋊C4 [×4], C2×C16 [×4], M5(2) [×8], C2×C42, C22×C8 [×2], C16⋊C4 [×4], C2×C8⋊C4, C2×M5(2) [×2], C2×C16⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C16⋊C4 [×2], C2×C8⋊C4, C2×C16⋊C4

Smallest permutation representation of C2×C16⋊C4
On 32 points
Generators in S32
(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)
(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 20)(2 25 10 17)(3 30)(4 19 12 27)(5 24)(6 29 14 21)(7 18)(8 23 16 31)(9 28)(11 22)(13 32)(15 26)

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

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

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

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

׿
×
𝔽