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G = C42⋊D4order 128 = 27

1st semidirect product of C42 and D4 acting faithfully

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

Aliases: C421D4, C24.27D4, C4.7(C4×D4), C41D44C4, C421(C2×C4), (C2×D4).81D4, C4.4D43C4, C4.9C4211C2, C23.134(C2×D4), C4.139(C4⋊D4), C22.11C244C2, C22.34C22≀C2, (C22×C4).35C23, C42⋊C2216C2, C23.15(C22⋊C4), C22.29C24.2C2, (C22×D4).33C22, C42⋊C2.33C22, C2.50(C23.23D4), (C2×M4(2)).194C22, C22.9(C22.D4), (C2×D4)⋊12(C2×C4), (C2×Q8)⋊12(C2×C4), (C2×C4).246(C2×D4), (C2×C4.D4)⋊20C2, (C2×C4).330(C4○D4), (C2×C4).195(C22×C4), (C2×C4○D4).29C22, C22.49(C2×C22⋊C4), SmallGroup(128,643)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42⋊D4
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C42⋊D4
C1C2C2×C4 — C42⋊D4
C1C2C22×C4 — C42⋊D4
C1C2C2C22×C4 — C42⋊D4

Generators and relations for C42⋊D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 444 in 181 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×7], C22 [×3], C22 [×18], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×16], Q8 [×2], C23, C23 [×4], C23 [×7], C42 [×6], C22⋊C4 [×11], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×4], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], C4.D4 [×2], C4≀C2 [×4], C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C2×M4(2) [×2], C22×D4, C2×C4○D4, C4.9C42 [×2], C2×C4.D4, C42⋊C22 [×2], C22.11C24, C22.29C24, C42⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42⋊D4

Character table of C42⋊D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11222444482222444444448888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-1-111111-1-1-1111-11-111-11-1    linear of order 2
ρ311111-1-1-1-1-11111-1111-1-1-111-11-11-11    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ511111-1-1-1-111111-1111-1-1-11-11-11-11-1    linear of order 2
ρ6111111111-11111-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ7111111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ811111-1-1-1-1111111-1-1-1111-1-11-1-11-11    linear of order 2
ρ911-11-11-11-1-1-111-1i-ii-i-ii-ii-111ii-i-i    linear of order 4
ρ1011-11-1-11-11-1-111-1-i-ii-ii-iii11-1i-i-ii    linear of order 4
ρ1111-11-11-11-11-111-1-ii-iii-ii-i1-1-1ii-i-i    linear of order 4
ρ1211-11-1-11-111-111-1ii-ii-ii-i-i-1-11i-i-ii    linear of order 4
ρ1311-11-1-11-11-1-111-1ii-ii-ii-i-i11-1-iii-i    linear of order 4
ρ1411-11-11-11-1-1-111-1-ii-iii-ii-i-111-i-iii    linear of order 4
ρ1511-11-1-11-111-111-1-i-ii-ii-iii-1-11-iii-i    linear of order 4
ρ1611-11-11-11-11-111-1i-ii-i-ii-ii1-1-1-i-iii    linear of order 4
ρ1722-2-22000002-22-22000-2-2200000000    orthogonal lifted from D4
ρ1822-22-2-2-22202-2-22000000000000000    orthogonal lifted from D4
ρ1922222-222-20-2-2-2-2000000000000000    orthogonal lifted from D4
ρ2022-2-22000002-22-2-200022-200000000    orthogonal lifted from D4
ρ2122-2-2200000-22-2202-2-200020000000    orthogonal lifted from D4
ρ2222-22-222-2-202-2-22000000000000000    orthogonal lifted from D4
ρ23222222-2-220-2-2-2-2000000000000000    orthogonal lifted from D4
ρ2422-2-2200000-22-220-222000-20000000    orthogonal lifted from D4
ρ25222-2-20000022-2-20-2i-2i2i0002i0000000    complex lifted from C4○D4
ρ26222-2-200000-2-2222i0002i-2i-2i00000000    complex lifted from C4○D4
ρ27222-2-20000022-2-202i2i-2i000-2i0000000    complex lifted from C4○D4
ρ28222-2-200000-2-222-2i000-2i2i2i00000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C42⋊D4
On 16 points - transitive group 16T222
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8 14 10)(2 5 15 11)(3 6 16 12)(4 7 13 9)
(1 3 14 16)(2 5)(4 9)(6 8 12 10)(7 13)(11 15)
(2 9)(3 16)(4 5)(6 12)(7 15)(11 13)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,14,10)(2,5,15,11)(3,6,16,12)(4,7,13,9), (1,3,14,16)(2,5)(4,9)(6,8,12,10)(7,13)(11,15), (2,9)(3,16)(4,5)(6,12)(7,15)(11,13)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8,14,10)(2,5,15,11)(3,6,16,12)(4,7,13,9), (1,3,14,16)(2,5)(4,9)(6,8,12,10)(7,13)(11,15), (2,9)(3,16)(4,5)(6,12)(7,15)(11,13) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8,14,10),(2,5,15,11),(3,6,16,12),(4,7,13,9)], [(1,3,14,16),(2,5),(4,9),(6,8,12,10),(7,13),(11,15)], [(2,9),(3,16),(4,5),(6,12),(7,15),(11,13)])

G:=TransitiveGroup(16,222);

On 16 points - transitive group 16T276
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 14 15 16)
(1 15 6 9)(2 13 5 11)(3 16 8 12)(4 14 7 10)
(1 9)(2 11)(3 10)(4 12)(5 13)(6 15)(7 16)(8 14)

G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,15,6,9)(2,13,5,11)(3,16,8,12)(4,14,7,10), (1,9)(2,11)(3,10)(4,12)(5,13)(6,15)(7,16)(8,14)>;

G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,15,6,9)(2,13,5,11)(3,16,8,12)(4,14,7,10), (1,9)(2,11)(3,10)(4,12)(5,13)(6,15)(7,16)(8,14) );

G=PermutationGroup([(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,14,15,16)], [(1,15,6,9),(2,13,5,11),(3,16,8,12),(4,14,7,10)], [(1,9),(2,11),(3,10),(4,12),(5,13),(6,15),(7,16),(8,14)])

G:=TransitiveGroup(16,276);

On 16 points - transitive group 16T305
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 5 3 8)(2 6 4 7)(9 16 11 14)(10 13 12 15)
(1 16)(2 10 4 12)(3 14)(5 9)(6 15 7 13)(8 11)
(1 11)(2 15)(3 9)(4 13)(5 14)(6 10)(7 12)(8 16)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,3,8)(2,6,4,7)(9,16,11,14)(10,13,12,15), (1,16)(2,10,4,12)(3,14)(5,9)(6,15,7,13)(8,11), (1,11)(2,15)(3,9)(4,13)(5,14)(6,10)(7,12)(8,16)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,5,3,8)(2,6,4,7)(9,16,11,14)(10,13,12,15), (1,16)(2,10,4,12)(3,14)(5,9)(6,15,7,13)(8,11), (1,11)(2,15)(3,9)(4,13)(5,14)(6,10)(7,12)(8,16) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,5,3,8),(2,6,4,7),(9,16,11,14),(10,13,12,15)], [(1,16),(2,10,4,12),(3,14),(5,9),(6,15,7,13),(8,11)], [(1,11),(2,15),(3,9),(4,13),(5,14),(6,10),(7,12),(8,16)])

G:=TransitiveGroup(16,305);

Matrix representation of C42⋊D4 in GL8(ℤ)

00001000
00000100
00000010
00000001
00010000
00-100000
0-1000000
10000000
,
01000000
-10000000
00010000
00-100000
00000100
0000-1000
00000001
000000-10
,
01000000
10000000
00010000
00100000
00000001
00000010
00000-100
0000-1000
,
10000000
01000000
00-100000
000-10000
00000010
00000001
00001000
00000100

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42⋊D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes D_4
% in TeX

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

G:=SmallGroup(128,643);
// by ID

G=gap.SmallGroup(128,643);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,2804,1411,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of C42⋊D4 in TeX

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