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G = (C2×C4).Q16order 128 = 27

1st non-split extension by C2×C4 of Q16 acting via Q16/C2=D4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: (C2×C4).1Q16, (C2×C4).5SD16, C22.19C4≀C2, (C22×C4).33D4, C2.C42.5C4, C22.C42.8C2, C22.60(C23⋊C4), C2.5(C42.C4), C2.7(C23.D4), C23.160(C22⋊C4), C22.18(Q8⋊C4), C2.6(C23.31D4), C23.83C23.1C2, C22.M4(2).5C2, (C2×C4⋊C4).9C4, (C2×C4⋊C4).7C22, (C22×C4).7(C2×C4), SmallGroup(128,85)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4).Q16
C1C2C22C23C22×C4C2×C4⋊C4C23.83C23 — (C2×C4).Q16
C1C2C23C22×C4 — (C2×C4).Q16
C1C22C23C2×C4⋊C4 — (C2×C4).Q16
C1C22C23C2×C4⋊C4 — (C2×C4).Q16

Generators and relations for (C2×C4).Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=ab2c4, cbc-1=ab=ba, cac-1=ab2, ad=da, dbd-1=ab-1, dcd-1=b-1c-1 >

2C2
2C2
2C22
2C22
2C4
2C4
4C4
4C4
8C4
8C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
4C2×C4
4C8
4C2×C4
4C8
4C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
8C8
2M4(2)
2C22×C4
2C22×C4
2M4(2)
4C2×C8
4C4⋊C4
4C2×C8
4C4⋊C4
4M4(2)
2C2.C42
2C22⋊C8
2C2×M4(2)
2C2.C42

Character table of (C2×C4).Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-11-1-1-111-111-1-1    linear of order 2
ρ3111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-11-1-11-1-11-1-111    linear of order 2
ρ5111111-1-1-1-1-1111-1-iiii-i-i-ii    linear of order 4
ρ6111111-1-1-1-11-11-11iii-i-i-ii-i    linear of order 4
ρ7111111-1-1-1-1-1111-1i-i-i-iiii-i    linear of order 4
ρ8111111-1-1-1-11-11-11-i-i-iiii-ii    linear of order 4
ρ92222222-22-200-20000000000    orthogonal lifted from D4
ρ10222222-22-2200-20000000000    orthogonal lifted from D4
ρ112-22-22-2-202000000200200-2-2    symplectic lifted from Q16, Schur index 2
ρ122-22-22-2-202000000-200-20022    symplectic lifted from Q16, Schur index 2
ρ132-22-2-2202i0-2i0000001+i-1-i01-i-1+i00    complex lifted from C4≀C2
ρ142-22-2-220-2i02i0000001-i-1+i01+i-1-i00    complex lifted from C4≀C2
ρ152-22-2-220-2i02i000000-1+i1-i0-1-i1+i00    complex lifted from C4≀C2
ρ162-22-22-220-2000000--200-200-2--2    complex lifted from SD16
ρ172-22-22-220-2000000-200--200--2-2    complex lifted from SD16
ρ182-22-2-2202i0-2i000000-1-i1+i0-1+i1-i00    complex lifted from C4≀C2
ρ194444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ2044-4-4000000-2i0002i00000000    complex lifted from C23.D4
ρ214-4-4400000002i0-2i000000000    complex lifted from C42.C4
ρ224-4-440000000-2i02i000000000    complex lifted from C42.C4
ρ2344-4-40000002i000-2i00000000    complex lifted from C23.D4

Smallest permutation representation of (C2×C4).Q16
On 32 points
Generators in S32
(1 29)(2 6)(3 31)(4 8)(5 25)(7 27)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)(26 30)(28 32)
(1 13 25 18)(2 23 26 10)(3 20 27 15)(4 12 28 17)(5 9 29 22)(6 19 30 14)(7 24 31 11)(8 16 32 21)
(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)
(2 12 26 17)(3 27)(4 14 28 19)(6 16 30 21)(7 31)(8 10 32 23)(9 13)(11 20)(15 24)(18 22)

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

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

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

Matrix representation of (C2×C4).Q16 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
0001201
,
1600000
0160000
0013000
000400
0000130
000044
,
7130000
400000
000010
0001212
000400
001511115
,
1040000
1370000
001000
0001600
0001212
00001616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,4,0,0,0,0,0,4],[7,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,15,0,0,0,12,4,11,0,0,1,1,0,11,0,0,0,2,0,5],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,16,12,0,0,0,0,0,1,16,0,0,0,0,2,16] >;

(C2×C4).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4).Q_{16}
% in TeX

G:=Group("(C2xC4).Q16");
// GroupNames label

G:=SmallGroup(128,85);
// by ID

G=gap.SmallGroup(128,85);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,1690,521,248,2804]);
// Polycyclic

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

Export

Subgroup lattice of (C2×C4).Q16 in TeX
Character table of (C2×C4).Q16 in TeX

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