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G = C4.10D4order 32 = 25

2nd non-split extension by C4 of D4 acting via D4/C22=C2

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

Aliases: C4.10D4, M4(2).1C2, (C2×C4).C4, (C2×Q8).1C2, (C2×C4).2C22, C22.4(C2×C4), C2.5(C22⋊C4), SmallGroup(32,8)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4.10D4
C1C2C4C2×C4C2×Q8 — C4.10D4
C1C2C22 — C4.10D4
C1C2C2×C4 — C4.10D4
C1C2C2C2×C4 — C4.10D4

Generators and relations for C4.10D4
 G = < a,b,c | a4=1, b4=a2, c2=bab-1=a-1, ac=ca, cbc-1=a-1b3 >

2C2
2C4
2C4
2C8
2C8
2Q8
2Q8

Character table of C4.10D4

 class 12A2B4A4B4C4D8A8B8C8D
 size 11222444444
ρ111111111111    trivial
ρ211111-1-11-11-1    linear of order 2
ρ31111111-1-1-1-1    linear of order 2
ρ411111-1-1-11-11    linear of order 2
ρ5111-1-1-11-i-iii    linear of order 4
ρ6111-1-11-1-iii-i    linear of order 4
ρ7111-1-1-11ii-i-i    linear of order 4
ρ8111-1-11-1i-i-ii    linear of order 4
ρ922-22-2000000    orthogonal lifted from D4
ρ1022-2-22000000    orthogonal lifted from D4
ρ114-4000000000    symplectic faithful, Schur index 2

Permutation representations of C4.10D4
On 16 points - transitive group 16T40
Generators in S16
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16 7 10 5 12 3 14)(2 9 4 15 6 13 8 11)

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

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

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

G:=TransitiveGroup(16,40);

Matrix representation of C4.10D4 in GL4(𝔽3) generated by

0002
0110
0120
1000
,
0100
0002
1001
0110
,
0120
2001
2000
0010
G:=sub<GL(4,GF(3))| [0,0,0,1,0,1,1,0,0,1,2,0,2,0,0,0],[0,0,1,0,1,0,0,1,0,0,0,1,0,2,1,0],[0,2,2,0,1,0,0,0,2,0,0,1,0,1,0,0] >;

C4.10D4 in GAP, Magma, Sage, TeX

C_4._{10}D_4
% in TeX

G:=Group("C4.10D4");
// GroupNames label

G:=SmallGroup(32,8);
// by ID

G=gap.SmallGroup(32,8);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,86,302,248,58]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^4=a^2,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;
// generators/relations

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