Copied to
clipboard

G = C4×C4.D4  order 128 = 27

Direct product of C4 and C4.D4

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

Aliases: C4×C4.D4, C42.421D4, C23.4C42, C4.105(C4×D4), C24.26(C2×C4), (C23×C4).10C4, M4(2)⋊11(C2×C4), (C4×M4(2))⋊20C2, C22.10(C2×C42), C4○2(C22.C42), C22.C42⋊27C2, (C22×C4).650C23, C23.178(C22×C4), (C2×C42).232C22, (C22×D4).445C22, C22.21(C42⋊C2), (C2×M4(2)).303C22, C2.4(M4(2).8C22), (C2×C4×D4).10C2, C2.13(C4×C22⋊C4), (C2×C4).1(C22×C4), C2.4(C2×C4.D4), (C2×D4).157(C2×C4), (C2×C4).42(C4○D4), (C2×C4).1299(C2×D4), (C2×C22⋊C4).25C4, (C2×C4⋊C4).744C22, (C22×C4).436(C2×C4), (C2×C4.D4).14C2, (C2×C4).397(C22⋊C4), (C2×C4)○(C22.C42), C22.115(C2×C22⋊C4), SmallGroup(128,487)

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

Derived series
C1 — C22 — C4×C4.D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — C4×C4.D4
Lower central
C1 — C2 — C22 — C4×C4.D4
Upper central
C1 — C2×C4 — C2×C42 — C4×C4.D4
Jennings
C1 — C2 — C2 — C22×C4 — C4×C4.D4

Generators and relations for C4×C4.D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 356 in 180 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C8⋊C4, C4.D4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C2×M4(2), C23×C4, C22×D4, C22.C42, C4×M4(2), C2×C4.D4, C2×C4×D4, C4×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C2×C4.D4, M4(2).8C22, C4×C4.D4

Smallest permutation representation of C4×C4.D4
►On 32 points
Generators in S32
(1 12 31 17)(2 13 32 18)(3 14 25 19)(4 15 26 20)(5 16 27 21)(6 9 28 22)(7 10 29 23)(8 11 30 24)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R8A···8P
order122222222244444···444448···8
size111122444411112···244444···4

44 irreducible representations

dim111111112244
type+++++++
imageC1C2C2C2C2C4C4C4D4C4○D4C4.D4M4(2).8C22
kernelC4×C4.D4C22.C42C4×M4(2)C2×C4.D4C2×C4×D4C4.D4C2×C22⋊C4C23×C4C42C2×C4C4C2
# reps1222116444422

Matrix representation of C4×C4.D4 ►in GL6(𝔽17)

400000
040000
001000
000100
000010
000001
,
1600000
0160000
0001600
001000
0000016
000010
,
040000
1300000
0000016
0000160
001000
0001600
,
040000
400000
0000016
000010
001000
000100

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

C4×C4.D4 in GAP, Magma, Sage, TeX 

C_4\times C_4.D_4
 
% in TeX
 
G:=Group("C4xC4.D4");
 
// GroupNames label
 
G:=SmallGroup(128,487);
 
// by ID
 
G=gap.SmallGroup(128,487);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁