Copied to
clipboard

G = C4210D4order 128 = 27

4th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4210D4, M4(2)⋊2D4, C24.30D4, (C2×Q8)⋊7D4, C4⋊C4.80D4, C4.3C22≀C2, (C2×D4).87D4, C426C48C2, C4.41(C4⋊D4), C4.31(C41D4), C23.579(C2×D4), C2.9(C232D4), C22.197C22≀C2, C2.24(D4.9D4), C23.38D427C2, C22.57(C4⋊D4), C22.29C24.4C2, (C2×C42).343C22, (C22×C4).710C23, (C22×D4).60C22, (C22×Q8).49C22, C42⋊C2.48C22, (C2×M4(2)).13C22, (C2×C4≀C2)⋊25C2, (C2×C4.D4)⋊2C2, (C2×C4.4D4)⋊2C2, (C2×C8.C22)⋊1C2, (C2×C4).74(C4○D4), (C2×C4).1025(C2×D4), (C2×C4○D4).45C22, SmallGroup(128,736)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C4210D4
C1C2C22C23C22×C4C22×D4C2×C4.4D4 — C4210D4
C1C2C22×C4 — C4210D4
C1C22C22×C4 — C4210D4
C1C2C2C22×C4 — C4210D4

Generators and relations for C4210D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b-1, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 480 in 197 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×4], C4 [×7], C22 [×3], C22 [×17], C8 [×3], C2×C4 [×6], C2×C4 [×13], D4 [×12], Q8 [×8], C23, C23 [×11], C42 [×2], C42 [×2], C22⋊C4 [×13], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×3], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×11], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24 [×2], C4.D4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×5], C41D4, C2×M4(2) [×2], C2×SD16, C2×Q16, C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C426C4, C2×C4.D4, C23.38D4, C2×C4≀C2, C2×C4.4D4, C22.29C24, C2×C8.C22, C4210D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.9D4 [×2], C4210D4

Character table of C4210D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112288822224444888888888
ρ111111111111111111111111111    trivial
ρ2111111-111111111111-1-1-11-1-1-1-1    linear of order 2
ρ31111111-1-111111111-11-1-1-1-1-111    linear of order 2
ρ4111111-1-1-111111111-1-111-111-1-1    linear of order 2
ρ51111111111111-1-1-1-1-1111-1-1-1-1-1    linear of order 2
ρ6111111-1111111-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ71111111-1-11111-1-1-1-111-1-1111-1-1    linear of order 2
ρ8111111-1-1-11111-1-1-1-11-1111-1-111    linear of order 2
ρ92-2-22-22000-222-200002000-20000    orthogonal lifted from D4
ρ102222-2-2000-22-22000000-2200000    orthogonal lifted from D4
ρ112222220-22-2-2-2-20000000000000    orthogonal lifted from D4
ρ122-2-22-220002-2-22-22-22000000000    orthogonal lifted from D4
ρ132222-2-2000-22-220000002-200000    orthogonal lifted from D4
ρ142-2-22-22000-222-20000-200020000    orthogonal lifted from D4
ρ152-2-222-2000-2-222000000000-2200    orthogonal lifted from D4
ρ162-2-222-2000-2-2220000000002-200    orthogonal lifted from D4
ρ172-2-22-220002-2-222-22-2000000000    orthogonal lifted from D4
ρ182222-2-2-2002-22-20000020000000    orthogonal lifted from D4
ρ192222-2-22002-22-200000-20000000    orthogonal lifted from D4
ρ2022222202-2-2-2-2-20000000000000    orthogonal lifted from D4
ρ212-2-222-200022-2-200000000000-2i2i    complex lifted from C4○D4
ρ222-2-222-200022-2-2000000000002i-2i    complex lifted from C4○D4
ρ234-44-40000000002i2i-2i-2i000000000    complex lifted from D4.9D4
ρ2444-4-4000000000-2i2i2i-2i000000000    complex lifted from D4.9D4
ρ2544-4-40000000002i-2i-2i2i000000000    complex lifted from D4.9D4
ρ264-44-4000000000-2i-2i2i2i000000000    complex lifted from D4.9D4

Smallest permutation representation of C4210D4
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 23 13 7)(2 24 14 8)(3 21 15 5)(4 22 16 6)(9 29 28 20)(10 30 25 17)(11 31 26 18)(12 32 27 19)
(1 26 15 28)(2 30 16 32)(3 9 13 11)(4 19 14 17)(5 20 23 18)(6 27 24 25)(7 31 21 29)(8 10 22 12)
(1 28)(2 30)(3 11)(4 19)(5 31)(6 12)(7 20)(8 25)(9 13)(10 24)(14 17)(15 26)(16 32)(18 21)(22 27)(23 29)

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

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

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

Matrix representation of C4210D4 in GL6(𝔽17)

0160000
1600000
0041300
00813134
0000130
0000013
,
1600000
0160000
0016101
0015111
000001
0000160
,
0160000
100000
00116016
00215115
0021600
0001161
,
1600000
010000
0016000
0015010
0015100
0001161

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,4,8,0,0,0,0,13,13,0,0,0,0,0,13,13,0,0,0,0,4,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,15,0,0,0,0,1,1,0,0,0,0,0,1,0,16,0,0,1,1,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,2,2,0,0,0,16,15,16,1,0,0,0,1,0,16,0,0,16,15,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,15,15,0,0,0,0,0,1,1,0,0,0,1,0,16,0,0,0,0,0,1] >;

C4210D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{10}D_4
% in TeX

G:=Group("C4^2:10D4");
// GroupNames label

G:=SmallGroup(128,736);
// by ID

G=gap.SmallGroup(128,736);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,2019,1018,521,248,4037]);
// 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^-1,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Character table of C4210D4 in TeX

׿
×
𝔽