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G = C8.26D4order 64 = 26

13rd non-split extension by C8 of D4 acting via D4/C22=C2

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

Aliases: D84C4, Q164C4, C8.26D4, SD162C4, C42.12C22, M4(2).12C22, C4≀C26C2, C8○D47C2, C8.6(C2×C4), C8⋊C43C2, C4○D8.3C2, D4.4(C2×C4), C2.19(C4×D4), C4.80(C2×D4), Q8.4(C2×C4), C8.C44C2, (C2×C4).80C23, (C2×C8).51C22, C4.16(C22×C4), C4○D4.8C22, C22.2(C4○D4), SmallGroup(64,125)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.26D4
C1C2C4C2×C4C2×C8C8○D4 — C8.26D4
C1C2C4 — C8.26D4
C1C4C2×C8 — C8.26D4
C1C2C2C2×C4 — C8.26D4

Generators and relations for C8.26D4
 G = < a,b,c | a8=b4=1, c2=a2, bab-1=cac-1=a5, cbc-1=a2b-1 >

2C2
4C2
4C2
2C4
2C4
2C22
2C22
4C4
2C2×C4
2D4
2C2×C4
2D4
2C2×C4
2C8
2C8
2C2×C8
2M4(2)
2M4(2)
2C2×C8

Character table of C8.26D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J
 size 1124411244442222444444
ρ11111111111111111111111    trivial
ρ2111-111111-111-1-1-1-1-1-11-11-1    linear of order 2
ρ31111-1111-11-1-11111-1-11-11-1    linear of order 2
ρ4111-1-1111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ51111-111111-11-1-1-1-1-11-1-1-11    linear of order 2
ρ6111-1-11111-1-1111111-1-11-1-1    linear of order 2
ρ711111111-111-1-1-1-1-11-1-11-1-1    linear of order 2
ρ8111-11111-1-11-11111-11-1-1-11    linear of order 2
ρ911-11-1-1-11-i-11i-iii-i-1-i-i1ii    linear of order 4
ρ1011-111-1-11i-1-1-i-iii-i1i-i-1i-i    linear of order 4
ρ1111-1-11-1-11-i1-1i-iii-i-1ii1-i-i    linear of order 4
ρ1211-1-1-1-1-11i11-i-iii-i1-ii-1-ii    linear of order 4
ρ1311-1-1-1-1-11-i11ii-i-ii1i-i-1i-i    linear of order 4
ρ1411-1-11-1-11i1-1-ii-i-ii-1-i-i1ii    linear of order 4
ρ1511-111-1-11-i-1-1ii-i-ii1-ii-1-ii    linear of order 4
ρ1611-11-1-1-11i-11-ii-i-ii-1ii1-i-i    linear of order 4
ρ1722-20022-20000-2-222000000    orthogonal lifted from D4
ρ1822-20022-2000022-2-2000000    orthogonal lifted from D4
ρ1922200-2-2-200002i-2i2i-2i000000    complex lifted from C4○D4
ρ2022200-2-2-20000-2i2i-2i2i000000    complex lifted from C4○D4
ρ214-40004i-4i000000000000000    complex faithful
ρ224-4000-4i4i000000000000000    complex faithful

Permutation representations of C8.26D4
On 16 points - transitive group 16T113
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)
(1 15 3 9 5 11 7 13)(2 12 4 14 6 16 8 10)

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

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

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

G:=TransitiveGroup(16,113);

C8.26D4 is a maximal subgroup of
C42.283C23  D811D4  D8.13D4  C8.5S4
 D8p⋊C4: D16⋊C4  D244C4  D2410C4  D4010C4  D4016C4  Q16⋊F5  D564C4  D5610C4 ...
 M4(2).D2p: M4(2).51D4  M4(2)○D8  M4(2).22D6  C24.54D4  M4(2).22D10  C40.50D4  M4(2).22D14  C56.50D4 ...
 C8p.(C2×C4): Q32⋊C4  D84Dic3  D84Dic5  D8⋊F5  SD162F5  D84Dic7 ...
C8.26D4 is a maximal quotient of
SD16⋊C8  Q165C8  D85C8  C89D8  C812SD16  C815SD16  C89Q16  C8⋊M4(2)  C8.M4(2)  C83M4(2)  D4.3C42  C8.5C42  M4(2).3Q8  C42.28Q8  SD162F5  Q16⋊F5
 M4(2).D2p: M4(2).42D4  M4(2).43D4  M4(2).24D4  M4(2).22D6  D2410C4  C24.54D4  M4(2).22D10  D4016C4 ...
 (Cp×D8)⋊C4: C42.116D4  D84Dic3  D84Dic5  D8⋊F5  D84Dic7 ...
 C42.D2p: C42.107D4  D244C4  D4010C4  D564C4 ...

Matrix representation of C8.26D4 in GL4(𝔽5) generated by

0020
0001
4000
0300
,
2000
0100
0030
0004
,
0004
0020
0400
2000
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,3,2,0,0,0,0,1,0,0],[2,0,0,0,0,1,0,0,0,0,3,0,0,0,0,4],[0,0,0,2,0,0,4,0,0,2,0,0,4,0,0,0] >;

C8.26D4 in GAP, Magma, Sage, TeX

C_8._{26}D_4
% in TeX

G:=Group("C8.26D4");
// GroupNames label

G:=SmallGroup(64,125);
// by ID

G=gap.SmallGroup(64,125);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b^-1>;
// generators/relations

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Subgroup lattice of C8.26D4 in TeX
Character table of C8.26D4 in TeX

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