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G = C42.426D4order 128 = 27

59th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.426D4, C8⋊C223C4, C4○D4.25D4, C4.118(C4×D4), C8.C223C4, C22.8(C4×D4), C426C44C2, (C2×D4).277D4, M4(2)⋊6(C2×C4), (C2×Q8).218D4, C22⋊C4.122D4, D4.8(C22⋊C4), C23.131(C2×D4), Q8.8(C22⋊C4), C4.190(C4⋊D4), M4(2)⋊4C47C2, C22.33C22≀C2, D8⋊C22.2C2, C22.3(C4⋊D4), C42⋊C2215C2, (C22×C4).690C23, C23.C236C2, (C2×C42).289C22, C4.12(C22.D4), C42⋊C2.270C22, C2.45(C23.23D4), (C2×M4(2)).191C22, (C4×C4○D4)⋊1C2, (C2×C4≀C2)⋊17C2, (C2×D4)⋊11(C2×C4), (C2×Q8)⋊11(C2×C4), C4○D4.15(C2×C4), C4.22(C2×C22⋊C4), (C2×C4).1008(C2×D4), (C2×C4).12(C22×C4), (C2×C4).759(C4○D4), (C2×C4○D4).264C22, SmallGroup(128,638)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.426D4
C1C2C22C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.426D4
C1C2C2×C4 — C42.426D4
C1C4C22×C4 — C42.426D4
C1C2C2C22×C4 — C42.426D4

Generators and relations for C42.426D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2b2c-1 >

Subgroups: 340 in 166 conjugacy classes, 54 normal (38 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×9], C22 [×3], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×19], D4 [×2], D4 [×9], Q8 [×2], Q8 [×3], C23, C23 [×2], C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×3], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C23⋊C4 [×2], C4≀C2 [×4], C2×C42, C2×C42, C42⋊C2 [×2], C42⋊C2, C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C4○D8 [×2], C8⋊C22 [×2], C8⋊C22, C8.C22 [×2], C8.C22, C2×C4○D4 [×2], C426C4, M4(2)⋊4C4, C23.C23, C2×C4≀C2, C42⋊C22, C4×C4○D4, D8⋊C22, C42.426D4
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, C42.426D4

Permutation representations of C42.426D4
On 16 points - transitive group 16T225
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 2 4)(5 7 6 8)(9 10 11 12)(13 14 15 16)
(1 13 8 10)(2 15 7 12)(3 14 5 11)(4 16 6 9)
(1 14 2 16)(3 15 4 13)(5 10 6 12)(7 11 8 9)

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

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

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

G:=TransitiveGroup(16,225);

On 16 points - transitive group 16T311
Generators in S16
(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 9)(2 11)(3 10)(4 12)(5 15 6 13)(7 16 8 14)
(1 15 2 13)(3 16 4 14)(5 11 6 9)(7 12 8 10)

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

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

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

G:=TransitiveGroup(16,311);

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C···4I4J···4Q4R4S4T8A8B8C8D
order12222222444···44···44448888
size11222448112···24···48888888

32 irreducible representations

dim11111111112222224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4C4D4D4D4D4D4C4○D4C42.426D4
kernelC42.426D4C426C4M4(2)⋊4C4C23.C23C2×C4≀C2C42⋊C22C4×C4○D4D8⋊C22C8⋊C22C8.C22C42C22⋊C4C2×D4C2×Q8C4○D4C2×C4C1
# reps11111111442211244

Matrix representation of C42.426D4 in GL4(𝔽5) generated by

3000
0400
0030
0004
,
3000
0300
0030
0003
,
0003
1000
0200
0010
,
0100
4000
0004
0010
G:=sub<GL(4,GF(5))| [3,0,0,0,0,4,0,0,0,0,3,0,0,0,0,4],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,1,0,0,0,0,2,0,0,0,0,1,3,0,0,0],[0,4,0,0,1,0,0,0,0,0,0,1,0,0,4,0] >;

C42.426D4 in GAP, Magma, Sage, TeX

C_4^2._{426}D_4
% in TeX

G:=Group("C4^2.426D4");
// GroupNames label

G:=SmallGroup(128,638);
// by ID

G=gap.SmallGroup(128,638);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,1411,718,172,2028,1027]);
// Polycyclic

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

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