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G = C4212D4order 128 = 27

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

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

Aliases: C4212D4, (C2×Q8).95D4, C428C47C2, (C2×D4).104D4, (C22×C4).83D4, C23.592(C2×D4), C4.147(C4⋊D4), C22.C4222C2, C22.222C22≀C2, C2.28(D4.9D4), C2.30(D4.8D4), C23.36D434C2, 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

C1C22×C4 — C4212D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.36D4 — C4212D4
C1C2C22×C4 — C4212D4
C1C22C22×C4 — C4212D4
C1C2C2C22×C4 — C4212D4

Generators and relations for C4212D4
 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 [×3], C2 [×4], C4 [×4], C4 [×7], C22 [×3], C22 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×10], Q8 [×4], C23, C23 [×2], C42 [×2], C42, C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C2.C42 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C2×C4⋊C4 [×2], C4⋊D4 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C2×C4○D4 [×2], C22.C42, C428C4, C23.36D4 [×2], C2×C4≀C2 [×2], C22.31C24, C4212D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.8D4, D4.9D4, C4212D4

Character table of C4212D4

 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 C4212D4
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 5 8 9)(2 6 7 10)(3 12 14 16)(4 11 13 15)(17 26 19 28)(18 27 20 25)(21 30 23 32)(22 31 24 29)
(1 13 2 3)(4 7 14 8)(5 11 6 16)(9 15 10 12)(17 30 25 24)(18 31 26 21)(19 32 27 22)(20 29 28 23)
(1 29)(2 23)(3 28)(4 18)(5 22)(6 32)(7 21)(8 31)(9 24)(10 30)(11 27)(12 17)(13 20)(14 26)(15 25)(16 19)

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

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

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

Matrix representation of C4212D4 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] >;

C4212D4 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 C4212D4 in TeX

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