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G = C4⋊C4.96D4  order 128 = 27

51st non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.96D4, (C2×C8).46D4, C4.67C22≀C2, (C2×D4).107D4, (C22×D8).4C2, C4.25(C4⋊D4), C4.C42⋊12C2, C4.69(C4.4D4), C2.19(D4.4D4), C23.275(C4○D4), C23.37D4⋊29C2, (C22×C4).726C23, (C22×C8).112C22, (C22×D4).82C22, C22.233(C4⋊D4), C42.6C22⋊18C2, C42⋊C2.58C22, C2.18(C23.10D4), (C2×M4(2)).227C22, C22.54(C22.D4), (C2×C4).80(C4○D4), (C2×C4).1042(C2×D4), (C2×C4.D4)⋊24C2, SmallGroup(128,777)

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

Derived series
C1 — C22×C4 — C4⋊C4.96D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.D4 — C4⋊C4.96D4
Lower central
C1 — C2 — C22×C4 — C4⋊C4.96D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.96D4
Jennings
C1 — C2 — C2 — C22×C4 — C4⋊C4.96D4

Generators and relations for C4⋊C4.96D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b-1, dbd=ab, dcd=a2c3 >

Subgroups: 408 in 148 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C24, C4.D4, D4⋊C4, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×D8, C22×D4, C4.C42, C2×C4.D4, C23.37D4, C42.6C22, C22×D8, C4⋊C4.96D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.4D4, C4⋊C4.96D4

Character table of C4⋊C4.96D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J
 size 11112288882222884444888888
ρ111111111111111111111111111    trivial
ρ2111111-1-1-1-11111111111-1-11-1-11    linear of order 2
ρ3111111-1-1-1-11111-1-1111111-111-1    linear of order 2
ρ411111111111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ51111111-11-11111-1-1-1-1-1-11-111-11    linear of order 2
ρ6111111-11-111111-1-1-1-1-1-1-111-111    linear of order 2
ρ7111111-11-11111111-1-1-1-11-1-11-1-1    linear of order 2
ρ81111111-11-1111111-1-1-1-1-11-1-11-1    linear of order 2
ρ92-2-22-220-202-22-22000000000000    orthogonal lifted from D4
ρ102-2-222-220-2022-2-2000000000000    orthogonal lifted from D4
ρ112222-2-20000-222-200-22-22000000    orthogonal lifted from D4
ρ122222-2-200002-2-222-20000000000    orthogonal lifted from D4
ρ132-2-22-22020-2-22-22000000000000    orthogonal lifted from D4
ρ142222-2-20000-222-2002-22-2000000    orthogonal lifted from D4
ρ152-2-222-2-202022-2-2000000000000    orthogonal lifted from D4
ρ162222-2-200002-2-22-220000000000    orthogonal lifted from D4
ρ172222220000-2-2-2-2000000002i00-2i    complex lifted from C4○D4
ρ182222220000-2-2-2-200000000-2i002i    complex lifted from C4○D4
ρ192-2-22-2200002-22-200000002i00-2i0    complex lifted from C4○D4
ρ202-2-22-2200002-22-20000000-2i002i0    complex lifted from C4○D4
ρ212-2-222-20000-2-222000000-2i002i00    complex lifted from C4○D4
ρ222-2-222-20000-2-2220000002i00-2i00    complex lifted from C4○D4
ρ2344-4-4000000000000-2√202√20000000    orthogonal lifted from D4.4D4
ρ244-44-400000000000002√20-2√2000000    orthogonal lifted from D4.4D4
ρ2544-4-40000000000002√20-2√20000000    orthogonal lifted from D4.4D4
ρ264-44-40000000000000-2√202√2000000    orthogonal lifted from D4.4D4

Smallest permutation representation of C4⋊C4.96D4
►On 32 points
Generators in S32
(1 27 5 31)(2 28 6 32)(3 29 7 25)(4 30 8 26)(9 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)

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

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

Matrix representation of C4⋊C4.96D4 ►in GL6(𝔽17)

1600000
0160000
001200
00161600
004401
00013160
,
1390000
040000
00100611
00120110
0033125
0003512
,
120000
16160000
00111100
003000
0005143
0012121414
,
100000
16160000
00111100
003600
0005143
00121233

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,4,0,0,0,2,16,4,13,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,9,4,0,0,0,0,0,0,10,12,3,0,0,0,0,0,3,3,0,0,6,11,12,5,0,0,11,0,5,12],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,3,0,12,0,0,11,0,5,12,0,0,0,0,14,14,0,0,0,0,3,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,11,3,0,12,0,0,11,6,5,12,0,0,0,0,14,3,0,0,0,0,3,3] >;
 

C4⋊C4.96D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{96}D_4
 
% in TeX
 
G:=Group("C4:C4.96D4");
 
// GroupNames label
 
G:=SmallGroup(128,777);
 
// by ID
 
G=gap.SmallGroup(128,777);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,718,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1,d*b*d=a*b,d*c*d=a^2*c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.96D4 in TeX

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