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G = C42⋊11D4  order 128 = 27

5th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊11D4, (C2×Q8).94D4, C42⋊9C4⋊5C2, (C2×D4).103D4, (C22×C4).82D4, C23.591(C2×D4), C2.34(D4⋊4D4), C4.146(C4⋊D4), C22.C42⋊21C2, C22.221C22≀C2, C23.36D4⋊33C2, C22.26(C4⋊D4), (C2×C42).362C22, (C22×C4).724C23, C2.28(D4.10D4), C4.21(C22.D4), C22.36(C4.4D4), C2.12(C23.10D4), (C2×M4(2)).225C22, C22.31C24.5C2, (C2×C4≀C2)⋊26C2, (C2×C4).259(C2×D4), (C2×C4).342(C4○D4), (C2×C4⋊C4).124C22, (C2×C4○D4).59C22, SmallGroup(128,771)

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

Derived series
C1 — C22×C4 — C42⋊11D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.36D4 — C42⋊11D4
Lower central
C1 — C2 — C22×C4 — C42⋊11D4
Upper central
C1 — C22 — C22×C4 — C42⋊11D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊11D4

Generators and relations for C42⋊11D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 352 in 153 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4≀C2, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×M4(2), C2×C4○D4, C22.C42, C42⋊9C4, C23.36D4, C2×C4≀C2, C22.31C24, C42⋊11D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4⋊4D4, D4.10D4, C42⋊11D4

Character table of C42⋊11D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ2111111-1-11111-1-1-1-1-1-11-11-11111    linear of order 2
ρ3111111-1111111111-11-1-1-1-1-1-111    linear of order 2
ρ41111111-11111-1-1-1-11-1-11-11-1-111    linear of order 2
ρ5111111111111-1-1-1-1111-11-1-1-1-1-1    linear of order 2
ρ6111111-1-111111111-1-11111-1-1-1-1    linear of order 2
ρ7111111-111111-1-1-1-1-11-11-1111-1-1    linear of order 2
ρ81111111-1111111111-1-1-1-1-111-1-1    linear of order 2
ρ92222-2-20222-2-200000-200000000    orthogonal lifted from D4
ρ1022222200-2-2-2-2000000-20200000    orthogonal lifted from D4
ρ112222-2-2-20-2-22200002000000000    orthogonal lifted from D4
ρ1222222200-2-2-2-200000020-200000    orthogonal lifted from D4
ρ132222-2-20-222-2-200000200000000    orthogonal lifted from D4
ρ142-2-22-22002-22-2-22-220000000000    orthogonal lifted from D4
ρ152222-2-220-2-2220000-2000000000    orthogonal lifted from D4
ρ162-2-22-22002-22-22-22-20000000000    orthogonal lifted from D4
ρ172-2-22-2200-22-220000000-2i02i0000    complex lifted from C4○D4
ρ182-2-22-2200-22-2200000002i0-2i0000    complex lifted from C4○D4
ρ192-2-222-2002-2-220000000000002i-2i    complex lifted from C4○D4
ρ202-2-222-200-222-200000000002i-2i00    complex lifted from C4○D4
ρ212-2-222-2002-2-22000000000000-2i2i    complex lifted from C4○D4
ρ222-2-222-200-222-20000000000-2i2i00    complex lifted from C4○D4
ρ234-44-40000000022-2-20000000000    orthogonal lifted from D4⋊4D4
ρ244-44-400000000-2-2220000000000    orthogonal lifted from D4⋊4D4
ρ2544-4-4000000002-2-220000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2644-4-400000000-222-20000000000    symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of C42⋊11D4
►On 32 points
Generators in S32
(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

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

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

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

Matrix representation of C42⋊11D4 ►in GL6(𝔽17)

010000
1600000
0013000
000400
000010
000001
,
100000
010000
0013000
000400
000040
0000013
,
040000
400000
000100
0016000
0000016
000010
,
100000
0160000
0000016
000010
000100
0016000

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

C42⋊11D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{11}D_4
 
% in TeX
 
G:=Group("C4^2:11D4");
 
// GroupNames label
 
G:=SmallGroup(128,771);
 
// by ID
 
G=gap.SmallGroup(128,771);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2804,718,172,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊11D4 in TeX

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