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G = C2×C8.C4order 64 = 26

Direct product of C2 and C8.C4

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

Aliases: C2×C8.C4, C23.9Q8, M4(2).9C22, C8.18(C2×C4), (C2×C8).10C4, C4.74(C2×D4), C4.16(C4⋊C4), (C2×C4).21Q8, C4(C8.C4), (C2×C4).148D4, C22.1(C2×Q8), (C2×C8).88C22, C4.28(C22×C4), (C22×C8).12C2, (C2×C4).71C23, C22.21(C4⋊C4), (C2×M4(2)).15C2, (C22×C4).116C22, C2.15(C2×C4⋊C4), (C2×C4).75(C2×C4), SmallGroup(64,110)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C8.C4
C1C2C4C2×C4C22×C4C22×C8 — C2×C8.C4
C1C2C4 — C2×C8.C4
C1C2×C4C22×C4 — C2×C8.C4
C1C2C2C2×C4 — C2×C8.C4

Generators and relations for C2×C8.C4
 G = < a,b,c | a2=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C2
2C22
2C22
2C8
2C8
2C8
2C8
2M4(2)
2C2×C8
2C2×C8
2M4(2)

Character table of C2×C8.C4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ21-11-11-1-111-1-1111-11-11-1-1-111-1-111-1    linear of order 2
ρ31-11-11-1-111-1-1111-11-11-1-11-1-111-1-11    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-11-1-111-1-11-1-11-11-1111-11-11-11-1    linear of order 2
ρ6111111111111-1-1-1-1-1-1-1-1-1-111-1-111    linear of order 2
ρ71-11-11-1-111-1-11-1-11-11-111-11-11-11-11    linear of order 2
ρ8111111111111-1-1-1-1-1-1-1-111-1-111-1-1    linear of order 2
ρ91-11-1-111-1-11-11-11-111-11-1-iii-ii-i-ii    linear of order 4
ρ101111-1-1-1-1-1-111-1111-1-1-11iiii-i-i-i-i    linear of order 4
ρ111-11-1-111-1-11-11-11-111-11-1i-i-ii-iii-i    linear of order 4
ρ121111-1-1-1-1-1-111-1111-1-1-11-i-i-i-iiiii    linear of order 4
ρ131-11-1-111-1-11-111-11-1-11-11i-ii-i-ii-ii    linear of order 4
ρ141111-1-1-1-1-1-1111-1-1-1111-1-i-iiiii-i-i    linear of order 4
ρ151-11-1-111-1-11-111-11-1-11-11-ii-iii-ii-i    linear of order 4
ρ161111-1-1-1-1-1-1111-1-1-1111-1ii-i-i-i-iii    linear of order 4
ρ172-22-2-22-222-22-20000000000000000    orthogonal lifted from D4
ρ182222-2-22222-2-20000000000000000    orthogonal lifted from D4
ρ192-22-22-22-2-222-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ20222222-2-2-2-2-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ212-2-22002i-2i2i-2i002--2-2-2-2-22--200000000    complex lifted from C8.C4
ρ2222-2-200-2i-2i2i2i002--2--2-22-2-2-200000000    complex lifted from C8.C4
ρ232-2-22002i-2i2i-2i00-2-2--2--222-2-200000000    complex lifted from C8.C4
ρ2422-2-200-2i-2i2i2i00-2-2-2--2-222--200000000    complex lifted from C8.C4
ρ252-2-2200-2i2i-2i2i002-2--2--2-2-22-200000000    complex lifted from C8.C4
ρ2622-2-2002i2i-2i-2i002-2-2--22-2-2--200000000    complex lifted from C8.C4
ρ272-2-2200-2i2i-2i2i00-2--2-2-222-2--200000000    complex lifted from C8.C4
ρ2822-2-2002i2i-2i-2i00-2--2--2-2-222-200000000    complex lifted from C8.C4

Smallest permutation representation of C2×C8.C4
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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 3 30 5 28 7 26)(2 31 4 29 6 27 8 25)(9 18 15 20 13 22 11 24)(10 17 16 19 14 21 12 23)

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

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

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

C2×C8.C4 is a maximal subgroup of
C8.8C42  C8.9C42  C8.11C42  C8.13C42  C8.2C42  M5(2).C4  C8.4C42  C8.14C42  C8.5C42  C8.6C42  C24.19Q8  C24.9Q8  (C2×C8).103D4  C8○D4⋊C4  C4○D4.4Q8  C4○D4.5Q8  C8.(C4⋊C4)  C42.324D4  C42.106D4  C24.10Q8  M4(2).42D4  M4(2).24D4  C42.62Q8  C42.28Q8  C42.430D4  M4(2).5Q8  M4(2).6Q8  M4(2).27D4  C42.326D4  C42.116D4  M4(2).31D4  M4(2).32D4  M4(2).33D4  M4(2).10D4  M4(2).11D4  M4(2).12D4  M4(2).13D4  C23.20SD16  C23.21SD16  M5(2)⋊3C4  M5(2).1C4  M4(2).29C23  M4(2)○D8  M4(2).10C23  (C8×D5).C4
C2×C8.C4 is a maximal quotient of
C42.42Q8  C88M4(2)  C87M4(2)  C42.43Q8  C42.92D4  C42.21Q8  C24.19Q8  C42.322D4  C42.324D4  C24.10Q8  C42.430D4  (C8×D5).C4

Matrix representation of C2×C8.C4 in GL3(𝔽17) generated by

1600
010
001
,
100
090
052
,
1300
012
0616
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,9,5,0,0,2],[13,0,0,0,1,6,0,2,16] >;

C2×C8.C4 in GAP, Magma, Sage, TeX

C_2\times C_8.C_4
% in TeX

G:=Group("C2xC8.C4");
// GroupNames label

G:=SmallGroup(64,110);
// by ID

G=gap.SmallGroup(64,110);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,963,117,88]);
// Polycyclic

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

Export

Subgroup lattice of C2×C8.C4 in TeX
Character table of C2×C8.C4 in TeX

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