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G = C4⋊C4.12D4  order 128 = 27

12nd non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.12D4, (C2×D4).16D4, (C2×Q8).16D4, (C22×C4).16D4, C23.526(C2×D4), C2.10(D4⋊D4), C22.24(C4○D8), C23.31D4⋊7C2, C4⋊D4.10C22, (C22×C4).15C23, C22.D8.1C2, C22.SD16.4C2, C22⋊Q8.10C22, C2.9(D4.10D4), C22.136C22≀C2, C23.47D4⋊26C2, C22⋊C8.115C22, C22.20(C8⋊C22), C2.6(C23.7D4), C23.81C23⋊1C2, C22.M4(2)⋊7C2, C22.33C24.2C2, C2.C42.22C22, (C2×C4).204(C2×D4), (C2×C4⋊C4).20C22, SmallGroup(128,341)

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

Derived series
C1 — C22×C4 — C4⋊C4.12D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.33C24 — C4⋊C4.12D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.12D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.12D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.12D4

Generators and relations for C4⋊C4.12D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2b2, bab-1=dad-1=a-1, cac-1=a-1b2, cbc-1=a-1b, dbd-1=ab, dcd-1=b2c3 >

Subgroups: 252 in 108 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, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C2.C42, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C22.M4(2), C22.SD16, C23.31D4, C23.81C23, C22.D8, C23.47D4, C22.33C24, C4⋊C4.12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8⋊C22, D4⋊D4, D4.10D4, C23.7D4, C4⋊C4.12D4

Character table of C4⋊C4.12D4

 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
ρ1622-2-2-220-2i002i00000000-√-2-√2√-2√2    complex lifted from C4○D8
ρ1722-2-2-2202i00-2i00000000-√-2√2√-2-√2    complex lifted from C4○D8
ρ1822-2-2-2202i00-2i00000000√-2-√2-√-2√2    complex lifted from C4○D8
ρ1944-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4400000002000000-20000    symplectic lifted from D4.10D4, Schur index 2
ρ214-4-440000000-200000020000    symplectic lifted from D4.10D4, Schur index 2
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

Smallest permutation representation of C4⋊C4.12D4
►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 16 23 22)(10 11 24 17)(12 19 18 13)(14 15 20 21)

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

Matrix representation of C4⋊C4.12D4 ►in GL8(𝔽17)

01000000
160000000
00010000
001600000
00000100
000016000
00000001
000000160
,
13412120000
441250000
551340000
512440000
00000071
000000110
0000101600
000016700
,
13412120000
13135120000
12124130000
512440000
00000071
000000110
000016700
00007100
,
001600000
00010000
160000000
01000000
00007100
000011000
000000167
00000071

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[13,4,5,5,0,0,0,0,4,4,5,12,0,0,0,0,12,12,13,4,0,0,0,0,12,5,4,4,0,0,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,0,16,7,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0],[13,13,12,5,0,0,0,0,4,13,12,12,0,0,0,0,12,5,4,4,0,0,0,0,12,12,13,4,0,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,0,7,1] >;
 

C4⋊C4.12D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{12}D_4
 
% in TeX
 
G:=Group("C4:C4.12D4");
 
// GroupNames label
 
G:=SmallGroup(128,341);
 
// by ID
 
G=gap.SmallGroup(128,341);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,184,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b^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=b^2*c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.12D4 in TeX

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