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G = C4⋊C4.96D4order 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.C4212C2, C4.69(C4.4D4), C2.19(D4.4D4), C23.275(C4○D4), C23.37D429C2, (C22×C4).726C23, (C22×C8).112C22, (C22×D4).82C22, C22.233(C4⋊D4), C42.6C2218C2, 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

C1C22×C4 — C4⋊C4.96D4
C1C2C22C23C22×C4C2×M4(2)C2×C4.D4 — C4⋊C4.96D4
C1C2C22×C4 — C4⋊C4.96D4
C1C22C22×C4 — C4⋊C4.96D4
C1C2C2C22×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 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C8 [×5], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×12], C23, C23 [×12], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], D8 [×8], C22×C4, C2×D4 [×4], C2×D4 [×10], C24 [×2], C4.D4 [×4], D4⋊C4 [×4], C4⋊C8 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×D8 [×6], C22×D4 [×2], C4.C42, C2×C4.D4 [×2], C23.37D4 [×2], C42.6C22, C22×D8, C4⋊C4.96D4
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, D4.4D4 [×2], 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
ρ2344-4-4000000000000-220220000000    orthogonal lifted from D4.4D4
ρ244-44-40000000000000220-22000000    orthogonal lifted from D4.4D4
ρ2544-4-4000000000000220-220000000    orthogonal lifted from D4.4D4
ρ264-44-40000000000000-22022000000    orthogonal lifted from D4.4D4

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

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

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

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

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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