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G = C4⋊C4.6D4  order 128 = 27

6th non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.6D4, (C2×D4).12D4, (C2×Q8).12D4, C2.12C2≀C22, Q8⋊D4.4C2, C22⋊Q16⋊2C2, (C22×C4).14D4, C4⋊D4.6C22, C23.520(C2×D4), C22⋊C8.3C22, (C22×C4).9C23, C22⋊Q8.6C22, C2.8(D4.7D4), C22.22(C4○D8), C22.SD16.2C2, C2.6(D4.10D4), C22.130C22≀C2, (C22×Q8).3C22, C23.31D4⋊12C2, C23.78C23⋊1C2, C22.14(C8.C22), C22.M4(2)⋊6C2, C2.C42.17C22, C22.33C24.1C2, (C2×C4).198(C2×D4), (C2×C4⋊C4).16C22, SmallGroup(128,335)

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

Derived series
C1 — C22×C4 — C4⋊C4.6D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.33C24 — C4⋊C4.6D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.6D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.6D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.6D4

Generators and relations for C4⋊C4.6D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, bab-1=dad-1=a-1, cac-1=a-1b2, cbc-1=a-1b, dbd-1=ab, dcd-1=c3 >

Subgroups: 292 in 122 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, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×SD16, C2×Q16, C22×Q8, C22.M4(2), C22.SD16, C23.31D4, C23.78C23, Q8⋊D4, C22⋊Q16, C22.33C24, C4⋊C4.6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8.C22, D4.7D4, D4.10D4, C2≀C22, C4⋊C4.6D4

Character table of C4⋊C4.6D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112284444888888888888
ρ111111111111111111111111    trivial
ρ21111111-111-1-1-1-1-1111-1-11-11    linear of order 2
ρ3111111-111111-11-11-111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111-111-11-1-1-1-11-111-11-1    linear of order 2
ρ611111111111-1111-11-1-1-1-1-1-1    linear of order 2
ρ7111111-1-111-111-11-1-1-11-11-11    linear of order 2
ρ8111111-11111-1-11-1-1-1-1-11111    linear of order 2
ρ92222-2-2-202-20000002000000    orthogonal lifted from D4
ρ102222-2-200-2200-20200000000    orthogonal lifted from D4
ρ112222220-2-2-2-2002000000000    orthogonal lifted from D4
ρ122222-2-2202-2000000-2000000    orthogonal lifted from D4
ρ1322222202-2-2200-2000000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-220-2i002i00000000-√2√-2√2-√-2    complex lifted from C4○D8
ρ1622-2-2-2202i00-2i00000000-√2-√-2√2√-2    complex lifted from C4○D8
ρ1722-2-2-2202i00-2i00000000√2√-2-√2-√-2    complex lifted from C4○D8
ρ1822-2-2-220-2i002i00000000√2-√-2-√2√-2    complex lifted from C4○D8
ρ194-44-40000000000020-200000    orthogonal lifted from C2≀C22
ρ204-44-400000000000-20200000    orthogonal lifted from C2≀C22
ρ214-4-4400000002000000-20000    symplectic lifted from D4.10D4, Schur index 2
ρ2244-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-440000000-200000020000    symplectic lifted from D4.10D4, Schur index 2

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

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

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

Matrix representation of C4⋊C4.6D4 ►in GL6(𝔽17)

1620000
1610000
000100
001000
001112
001616016
,
7100000
12100000
000010
001112
0016000
000161616
,
7100000
1200000
000010
0016161615
0001600
001101
,
400000
4130000
001000
000100
001112
001616016

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,1,1,16,0,0,1,0,1,16,0,0,0,0,1,0,0,0,0,0,2,16],[7,12,0,0,0,0,10,10,0,0,0,0,0,0,0,1,16,0,0,0,0,1,0,16,0,0,1,1,0,16,0,0,0,2,0,16],[7,12,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,1,0,0,0,16,16,1,0,0,1,16,0,0,0,0,0,15,0,1],[4,4,0,0,0,0,0,13,0,0,0,0,0,0,1,0,1,16,0,0,0,1,1,16,0,0,0,0,1,0,0,0,0,0,2,16] >;
 

C4⋊C4.6D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._6D_4
 
% in TeX
 
G:=Group("C4:C4.6D4");
 
// GroupNames label
 
G:=SmallGroup(128,335);
 
// by ID
 
G=gap.SmallGroup(128,335);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,184,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.6D4 in TeX

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