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G = C4.D8order 64 = 26

1st non-split extension by C4 of D8 acting via D8/D4=C2

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

Aliases: C4.9D8, C4.11SD16, C42.3C22, C4⋊C82C2, (C2×D4).2C4, C41D4.1C2, (C2×C4).108D4, C2.4(D4⋊C4), C2.4(C4.D4), C22.39(C22⋊C4), (C2×C4).12(C2×C4), SmallGroup(64,12)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.D8
C1C2C22C2×C4C42C41D4 — C4.D8
C1C22C2×C4 — C4.D8
C1C22C42 — C4.D8
C1C22C22C42 — C4.D8

Generators and relations for C4.D8
 G = < a,b,c | a4=b8=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

8C2
8C2
2C4
4C22
4C22
4C22
4C22
4C22
4C22
2C23
2C23
4D4
4D4
4D4
4D4
4D4
4C8
4C8
4D4
2C2×C8
2C2×D4
2C2×D4
2C2×C8

Character table of C4.D8

 class 12A2B2C2D2E4A4B4C4D4E8A8B8C8D8E8F8G8H
 size 1111882222444444444
ρ11111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111111-111-1-111-1    linear of order 2
ρ41111-1-1111111-1-111-1-11    linear of order 2
ρ51111-11-1-1-1-11-i-iii-ii-ii    linear of order 4
ρ61111-11-1-1-1-11ii-i-ii-ii-i    linear of order 4
ρ711111-1-1-1-1-11i-ii-iii-i-i    linear of order 4
ρ811111-1-1-1-1-11-ii-ii-i-iii    linear of order 4
ρ92222002-2-22-200000000    orthogonal lifted from D4
ρ10222200-222-2-200000000    orthogonal lifted from D4
ρ112-2-2200-2002002200-2-20    orthogonal lifted from D8
ρ1222-2-20002-2002002-200-2    orthogonal lifted from D8
ρ132-2-2200-200200-2-200220    orthogonal lifted from D8
ρ1422-2-20002-200-200-22002    orthogonal lifted from D8
ρ1522-2-2000-2200--200-2-200--2    complex lifted from SD16
ρ162-2-2200200-200-2--200-2--20    complex lifted from SD16
ρ172-2-2200200-200--2-200--2-20    complex lifted from SD16
ρ1822-2-2000-2200-200--2--200-2    complex lifted from SD16
ρ194-44-4000000000000000    orthogonal lifted from C4.D4

Smallest permutation representation of C4.D8
On 32 points
Generators in S32
(1 9 22 25)(2 26 23 10)(3 11 24 27)(4 28 17 12)(5 13 18 29)(6 30 19 14)(7 15 20 31)(8 32 21 16)
(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 9 21 22 16 25 8)(2 7 26 15 23 20 10 31)(3 30 11 19 24 14 27 6)(4 5 28 13 17 18 12 29)

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

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

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

C4.D8 is a maximal subgroup of
D4⋊D8  C42.181C23  Q8⋊D8  D42SD16  C42.191C23  Q82SD16  D4.D8  C42.201C23  Q8.D8  Q83SD16  C88D8  C814SD16  C87D8  C813SD16  D4.2SD16  Q8.2SD16  D4.2D8  Q8.2D8  C8⋊D8  C8⋊SD16  C82D8  C82SD16  C42.248C23  C42.249C23  C42.252C23  C42.253C23  Dic5.SD16
 C42.D2p: C42.D4  C42.409D4  C42.411D4  C42.413D4  C42.78D4  C42.80D4  C42.417D4  C42.82D4 ...
C4.D8 is a maximal quotient of
(C2×C4).98D8  C42.8Q8  Dic5.SD16
 C4p.D8: C8.24D8  C8.25D8  C8.29D8  C8.30D8  C4.D16  C8.27D8  C4.D24  C12.9D8 ...

Matrix representation of C4.D8 in GL4(𝔽17) generated by

161500
1100
0010
0001
,
7700
51000
0066
00140
,
7700
5000
0066
001411
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[7,5,0,0,7,10,0,0,0,0,6,14,0,0,6,0],[7,5,0,0,7,0,0,0,0,0,6,14,0,0,6,11] >;

C4.D8 in GAP, Magma, Sage, TeX

C_4.D_8
% in TeX

G:=Group("C4.D8");
// GroupNames label

G:=SmallGroup(64,12);
// by ID

G=gap.SmallGroup(64,12);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,165]);
// Polycyclic

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

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Subgroup lattice of C4.D8 in TeX
Character table of C4.D8 in TeX

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