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G = C4⋊C4.20D4  order 128 = 27

20th non-split extension by C4⋊C4 of D4 acting faithfully

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

Aliases: C4⋊C4.20D4, (C2×C4).3Q16, (C2×Q8).21D4, (C22×C4).55D4, C23.534(C2×D4), C22⋊C8.9C22, C22.14(C2×Q16), (C22×C4).23C23, C2.9(C22⋊Q16), C22⋊Q8.15C22, C22.144C22≀C2, C2.11(D4.8D4), C23.31D4.3C2, C23.48D4.1C2, C2.14(C23.7D4), C22.34(C8.C22), C2.C42.29C22, C23.83C23.2C2, C23.41C23.3C2, C22.M4(2).8C2, (C2×C4).212(C2×D4), (C2×C4⋊C4).27C22, SmallGroup(128,349)

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

Derived series
C1 — C22×C4 — C4⋊C4.20D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.41C23 — C4⋊C4.20D4
Lower central
C1 — C22 — C22×C4 — C4⋊C4.20D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.20D4
Jennings
C1 — C2 — C22 — C22×C4 — C4⋊C4.20D4

Generators and relations for C4⋊C4.20D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, bab-1=dad-1=a-1, cac-1=ab2, cbc-1=dbd-1=a-1b-1, dcd-1=a2c-1 >

Subgroups: 220 in 101 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22.M4(2), C23.31D4, C23.83C23, C23.48D4, C23.41C23, C4⋊C4.20D4
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C8.C22, C22⋊Q16, D4.8D4, C23.7D4, C4⋊C4.20D4

Character table of C4⋊C4.20D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ2111111-1-111-11-11-11-11-1-11-11    linear of order 2
ρ311111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111111-11111-11-1-1-1-1-1-1    linear of order 2
ρ6111111-1-11111-11-1-1-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-1-1-11-1-11111    linear of order 2
ρ8111111-1-1111-11-11-1-1-11-11-11    linear of order 2
ρ92222-2-200-2200-20200000000    orthogonal lifted from D4
ρ1022222222-2-2000000-2000000    orthogonal lifted from D4
ρ112222-2-2002-2020-2000000000    orthogonal lifted from D4
ρ122222-2-2002-20-202000000000    orthogonal lifted from D4
ρ13222222-2-2-2-20000002000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-22-2200000000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ1622-2-2-222-200000000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-22-2200000000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1822-2-2-222-200000000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ1944-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-4-440000002i0000000-2i0000    complex lifted from D4.8D4
ρ214-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ224-44-400000000000-2i02i00000    complex lifted from C23.7D4
ρ234-4-44000000-2i00000002i0000    complex lifted from D4.8D4

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

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

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

Matrix representation of C4⋊C4.20D4 ►in GL6(𝔽17)

12120000
1250000
0016200
000100
00016016
00016160
,
1550000
1620000
00160150
0000161
001010
0011610
,
0160000
100000
001000
0011600
000001
00160160
,
0160000
100000
0013000
0001300
0000013
0000130

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

C4⋊C4.20D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{20}D_4
 
% in TeX
 
G:=Group("C4:C4.20D4");
 
// GroupNames label
 
G:=SmallGroup(128,349);
 
// by ID
 
G=gap.SmallGroup(128,349);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,232,422,352,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^-1*b^-1,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of C4⋊C4.20D4 in TeX

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