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G = C4⋊C4.D4  order 128 = 27

1st non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.1D4, (C2×D4).9D4, (C2×Q8).9D4, C2.6C2≀C22, C22⋊D8.1C2, (C22×C4).12D4, C22⋊SD16⋊26C2, C2.8(D4⋊D4), C23.514(C2×D4), C4⋊D4.2C22, (C22×C4).3C23, C22⋊Q8.2C22, C2.6(D4.8D4), C22.20(C4○D8), C23.10D4⋊1C2, C23.31D4⋊4C2, C22.SD16⋊10C2, (C22×D4).4C22, C22.124C22≀C2, C22⋊C8.110C22, C22.18(C8⋊C22), C22.33C24⋊1C2, C22.M4(2)⋊5C2, C2.C42.12C22, (C2×C4).192(C2×D4), (C2×C4⋊C4).13C22, SmallGroup(128,329)

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

Derived series
C1 — C22×C4 — C4⋊C4.D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.33C24 — C4⋊C4.D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.D4

Generators and relations for C4⋊C4.D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, bab-1=dad=a-1, cac-1=a-1b2, cbc-1=a-1b, dbd=ab, dcd=a2c3 >

Subgroups: 380 in 136 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×D8, C2×SD16, C22×D4, C22.M4(2), C22.SD16, C23.31D4, C23.10D4, C22⋊D8, C22⋊SD16, C22.33C24, C4⋊C4.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8⋊C22, D4⋊D4, D4.8D4, C2≀C22, C4⋊C4.D4

Character table of C4⋊C4.D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 11112288844448888888888
ρ111111111111111111111111    trivial
ρ21111111-11-1-111-11-11-1-11-11-1    linear of order 2
ρ3111111-11-11111-11111-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-1-11111-11-11-11-11    linear of order 2
ρ51111111-1111111-1-1-111-1-1-1-1    linear of order 2
ρ6111111111-1-111-1-11-1-1-1-11-11    linear of order 2
ρ7111111-1-1-11111-1-1-1-11-11111    linear of order 2
ρ8111111-11-1-1-1111-11-1-111-11-1    linear of order 2
ρ92222-2-200000-220-202000000    orthogonal lifted from D4
ρ10222222000-2-2-2-20000200000    orthogonal lifted from D4
ρ112222-2-2020002-200-20000000    orthogonal lifted from D4
ρ1222222200022-2-20000-200000    orthogonal lifted from D4
ρ132222-2-200000-22020-2000000    orthogonal lifted from D4
ρ142222-2-20-20002-20020000000    orthogonal lifted from D4
ρ1522-2-2-22000-2i2i00000000√-2-√2-√-2√2    complex lifted from C4○D8
ρ1622-2-2-22000-2i2i00000000-√-2√2√-2-√2    complex lifted from C4○D8
ρ1722-2-2-220002i-2i00000000√-2√2-√-2-√2    complex lifted from C4○D8
ρ1822-2-2-220002i-2i00000000-√-2-√2√-2√2    complex lifted from C4○D8
ρ194-44-40020-200000000000000    orthogonal lifted from C2≀C22
ρ204-44-400-20200000000000000    orthogonal lifted from C2≀C22
ρ2144-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-440000000002i0000-2i0000    complex lifted from D4.8D4
ρ234-4-44000000000-2i00002i0000    complex lifted from D4.8D4

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

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

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

550000
5120000
0001600
0016000
0016161615
001101
,
210000
12150000
000010
001112
0016000
000161616
,
520000
210000
000010
0016161615
000100
000001
,
040000
1300000
001000
000100
0016161615
000001

G:=sub<GL(6,GF(17))| [5,5,0,0,0,0,5,12,0,0,0,0,0,0,0,16,16,1,0,0,16,0,16,1,0,0,0,0,16,0,0,0,0,0,15,1],[2,12,0,0,0,0,1,15,0,0,0,0,0,0,0,1,16,0,0,0,0,1,0,16,0,0,1,1,0,16,0,0,0,2,0,16],[5,2,0,0,0,0,2,1,0,0,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,1,16,0,0,0,0,0,15,0,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,1,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,15,1] >;
 

C4⋊C4.D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4.D_4
 
% in TeX
 
G:=Group("C4:C4.D4");
 
// GroupNames label
 
G:=SmallGroup(128,329);
 
// by ID
 
G=gap.SmallGroup(128,329);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,184,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=a^2*c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.D4 in TeX

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