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G = C4211D4order 128 = 27

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

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

Aliases: C4211D4, (C2×Q8).94D4, C429C45C2, (C2×D4).103D4, (C22×C4).82D4, C23.591(C2×D4), C2.34(D44D4), C4.146(C4⋊D4), C22.C4221C2, C22.221C22≀C2, C23.36D433C2, 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

C1C22×C4 — C4211D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.36D4 — C4211D4
C1C2C22×C4 — C4211D4
C1C22C22×C4 — C4211D4
C1C2C2C22×C4 — C4211D4

Generators and relations for C4211D4
 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 [×3], C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×10], Q8 [×4], C23, C23 [×2], C42 [×2], C42, C22⋊C4 [×4], C4⋊C4 [×10], 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], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C2×C4○D4 [×2], C22.C42, C429C4, C23.36D4 [×2], C2×C4≀C2 [×2], C22.31C24, C4211D4
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, D44D4, D4.10D4, C4211D4

Character table of C4211D4

 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 D44D4
ρ244-44-400000000-2-2220000000000    orthogonal lifted from D44D4
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 C4211D4
On 32 points
Generators in S32
(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 3 8 6)(2 5 7 4)(9 11 12 16)(10 14 15 13)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 16 5 15)(2 13 6 9)(3 12 7 14)(4 10 8 11)(17 30 21 25)(18 29 22 28)(19 32 23 27)(20 31 24 26)
(1 28)(2 32)(3 25)(4 31)(5 29)(6 27)(7 30)(8 26)(9 23)(10 20)(11 24)(12 21)(13 19)(14 17)(15 18)(16 22)

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

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

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

Matrix representation of C4211D4 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] >;

C4211D4 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 C4211D4 in TeX

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