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G = C42.82D4  order 128 = 27

64th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.82D4, C42.173C23, C4⋊C8⋊47C22, C4.D8⋊24C2, C4⋊1D4.15C4, C42.114(C2×C4), (C22×D4).12C4, (C22×C4).242D4, C4.111(C8⋊C22), C42.6C4⋊42C2, C4⋊1D4.130C22, (C2×C42).217C22, C23.185(C22⋊C4), C22.18(C4.D4), C2.14(C23.37D4), (C2×D4).34(C2×C4), (C2×C4⋊1D4).3C2, (C2×C4).1244(C2×D4), C2.21(C2×C4.D4), (C22×C4).239(C2×C4), (C2×C4).167(C22×C4), (C2×C4).106(C22⋊C4), C22.231(C2×C22⋊C4), SmallGroup(128,287)

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

Derived series
C1 — C2×C4 — C42.82D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊1D4 — C42.82D4
Lower central
C1 — C22 — C2×C4 — C42.82D4
Upper central
C1 — C22 — C2×C42 — C42.82D4
Jennings
C1 — C22 — C22 — C42 — C42.82D4

Generators and relations for C42.82D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 484 in 166 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C42, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4⋊1D4, C4⋊1D4, C22×D4, C22×D4, C4.D8, C42.6C4, C2×C4⋊1D4, C42.82D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C8⋊C22, C2×C4.D4, C23.37D4, C42.82D4

Character table of C42.82D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11112288882222444488888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-11-11111-11-1-11-11-1-11-11    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-11-11-11111-11-1-1-11-111-11-1    linear of order 2
ρ5111111-1-1-1-111111111-1111-1-1-11    linear of order 2
ρ61111-1-1-11-111111-11-1-1-1-11-11-111    linear of order 2
ρ7111111-1-1-1-1111111111-1-1-1111-1    linear of order 2
ρ81111-1-1-11-111111-11-1-111-11-11-1-1    linear of order 2
ρ9111111-1-111-1-1-1-1-11-11ii-i-ii-i-ii    linear of order 4
ρ101111-1-1-111-1-1-1-1-1111-1i-i-ii-i-iii    linear of order 4
ρ11111111-1-111-1-1-1-1-11-11-i-iii-iii-i    linear of order 4
ρ121111-1-1-111-1-1-1-1-1111-1-iii-iii-i-i    linear of order 4
ρ1311111111-1-1-1-1-1-1-11-11-ii-i-i-iiii    linear of order 4
ρ141111-1-11-1-11-1-1-1-1111-1-i-i-iiii-ii    linear of order 4
ρ1511111111-1-1-1-1-1-1-11-11i-iiii-i-i-i    linear of order 4
ρ161111-1-11-1-11-1-1-1-1111-1iii-i-i-ii-i    linear of order 4
ρ172222-2-20000-2-2222-2-2200000000    orthogonal lifted from D4
ρ18222222000022-2-22-2-2-200000000    orthogonal lifted from D4
ρ192222-2-2000022-2-2-2-22200000000    orthogonal lifted from D4
ρ202222220000-2-222-2-22-200000000    orthogonal lifted from D4
ρ214-44-40000004-400000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4-4400000000000000000000    orthogonal lifted from C4.D4
ρ234-4-4400000000-44000000000000    orthogonal lifted from C8⋊C22
ρ244-44-4000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-44-400000000000000000000    orthogonal lifted from C4.D4
ρ264-4-44000000004-4000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C42.82D4
►On 32 points
Generators in S32
(1 32 18 9)(2 29 19 14)(3 26 20 11)(4 31 21 16)(5 28 22 13)(6 25 23 10)(7 30 24 15)(8 27 17 12)

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

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

Matrix representation of C42.82D4 ►in GL8(𝔽17)

00100000
00010000
10000000
01000000
00000100
000016000
0000215016
00002210
,
01000000
160000000
00010000
001600000
00000100
000016000
00002201
0000215160
,
222150000
21515150000
15215150000
221520000
0000013015
0000013150
00000040
00009840
,
21515150000
222150000
221520000
15215150000
0000013150
0000013015
000001604
00009804

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,2,2,0,0,0,0,1,0,15,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,2,2,0,0,0,0,1,0,2,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[2,2,15,2,0,0,0,0,2,15,2,2,0,0,0,0,2,15,15,15,0,0,0,0,15,15,15,2,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,13,13,0,8,0,0,0,0,0,15,4,4,0,0,0,0,15,0,0,0],[2,2,2,15,0,0,0,0,15,2,2,2,0,0,0,0,15,2,15,15,0,0,0,0,15,15,2,15,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,13,13,16,8,0,0,0,0,15,0,0,0,0,0,0,0,0,15,4,4] >;
 

C42.82D4 in GAP, Magma, Sage, TeX 

C_4^2._{82}D_4
 
% in TeX
 
G:=Group("C4^2.82D4");
 
// GroupNames label
 
G:=SmallGroup(128,287);
 
// by ID
 
G=gap.SmallGroup(128,287);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,1018,248,1971,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
 
// generators/relations
 

Export

Character table of C42.82D4 in TeX

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