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G = C42⋊12D4  order 128 = 27

6th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊12D4, (C2×Q8).95D4, C42⋊8C4⋊7C2, (C2×D4).104D4, (C22×C4).83D4, C23.592(C2×D4), C4.147(C4⋊D4), C22.C42⋊22C2, C22.222C22≀C2, C2.28(D4.9D4), C2.30(D4.8D4), C23.36D4⋊34C2, C22.27(C4⋊D4), (C22×C4).725C23, (C2×C42).363C22, C4.22(C22.D4), C22.37(C4.4D4), C2.13(C23.10D4), (C2×M4(2)).226C22, C22.31C24.6C2, (C2×C4≀C2)⋊27C2, (C2×C4).260(C2×D4), (C2×C4).343(C4○D4), (C2×C4⋊C4).125C22, (C2×C4○D4).60C22, SmallGroup(128,772)

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

Derived series
C1 — C22×C4 — C42⋊12D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.36D4 — C42⋊12D4
Lower central
C1 — C2 — C22×C4 — C42⋊12D4
Upper central
C1 — C22 — C22×C4 — C42⋊12D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊12D4

Generators and relations for C42⋊12D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 336 in 145 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, C2.C42, D4⋊C4, Q8⋊C4, C4≀C2, C2×C42, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×M4(2), C2×C4○D4, C22.C42, C42⋊8C4, C23.36D4, C2×C4≀C2, C22.31C24, C42⋊12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.8D4, D4.9D4, C42⋊12D4

Character table of C42⋊12D4

 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
ρ234-44-4000000002i2i-2i-2i0000000000    complex lifted from D4.9D4
ρ244-44-400000000-2i-2i2i2i0000000000    complex lifted from D4.9D4
ρ2544-4-4000000002i-2i-2i2i0000000000    complex lifted from D4.8D4
ρ2644-4-400000000-2i2i2i-2i0000000000    complex lifted from D4.8D4

Smallest permutation representation of C42⋊12D4
►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 9 4 7)(2 10 3 8)(5 11 14 16)(6 12 13 15)(17 32 19 30)(18 29 20 31)(21 26 23 28)(22 27 24 25)

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

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

Matrix representation of C42⋊12D4 ►in GL6(𝔽17)

010000
1600000
0013000
0001300
0000160
000001
,
1600000
0160000
004000
0001300
0000130
000004
,
040000
400000
0001600
001000
000001
0000160
,
040000
1300000
000001
0000160
0001600
001000

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

C42⋊12D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{12}D_4
 
% in TeX
 
G:=Group("C4^2:12D4");
 
// GroupNames label
 
G:=SmallGroup(128,772);
 
// by ID
 
G=gap.SmallGroup(128,772);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,394,2804,718,172,2028,1027,124]);
 
// 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*b^2,d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C42⋊12D4 in TeX

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