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G = C2×C42.3C4  order 128 = 27

Direct product of C2 and C42.3C4

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

Aliases: C2×C42.3C4, C4⋊Q8.27C4, (C2×C42).24C4, C42.26(C2×C4), (C2×Q8).118D4, (C22×C4).98D4, C4⋊Q8.252C22, (C22×Q8).13C4, (C2×Q8).11C23, C22.53(C23⋊C4), C4.10D4.5C22, (C22×Q8).85C22, C23.204(C22⋊C4), (C2×C4).8(C2×D4), (C2×C4⋊Q8).23C2, C2.43(C2×C23⋊C4), (C2×Q8).38(C2×C4), (C22×C4).83(C2×C4), (C2×C4).30(C22⋊C4), (C2×C4).100(C22×C4), C22.67(C2×C22⋊C4), (C2×C4.10D4).13C2, SmallGroup(128,863)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×C42.3C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C2×C4⋊Q8 — C2×C42.3C4
Lower central
C1 — C2 — C22 — C2×C4 — C2×C42.3C4
Upper central
C1 — C22 — C23 — C22×Q8 — C2×C42.3C4
Jennings
C1 — C2 — C22 — C2×Q8 — C2×C42.3C4

Generators and relations for C2×C42.3C4
 G = < a,b,c,d | a2=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 244 in 120 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C4.10D4, C4.10D4, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C42.3C4, C2×C4.10D4, C2×C4⋊Q8, C2×C42.3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C42.3C4, C2×C23⋊C4, C2×C42.3C4

Character table of C2×C42.3C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111-11-1-11111-11-1-1-11-1-111-11    linear of order 2
ρ31-11-11-11-1-11111-1-1-11-1-1111-11-1-1    linear of order 2
ρ41-11-11-1-1-11-1111-11-1-1111-1-1-111-1    linear of order 2
ρ51-11-11-11-1-11111-1-1-11-11-1-1-11-111    linear of order 2
ρ61-11-11-1-1-11-1111-11-1-11-1-1111-1-11    linear of order 2
ρ7111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111-11-1-11111-11-1-11-111-1-11-1    linear of order 2
ρ9111111-1-1-1-1-11-1-1-1111i-ii-iii-i-i    linear of order 4
ρ101-11-11-1-111-1-11-111-11-1-i-ii-i-iiii    linear of order 4
ρ111111111-111-11-1-111-1-1-i-i-iiiii-i    linear of order 4
ρ121-11-11-111-11-11-11-1-1-11i-i-ii-ii-ii    linear of order 4
ρ13111111-1-1-1-1-11-1-1-1111-ii-ii-i-iii    linear of order 4
ρ141-11-11-1-111-1-11-111-11-1ii-iii-i-i-i    linear of order 4
ρ151111111-111-11-1-111-1-1iii-i-i-i-ii    linear of order 4
ρ161-11-11-111-11-11-11-1-1-11-iii-ii-ii-i    linear of order 4
ρ172222220-2002-2-220-20000000000    orthogonal lifted from D4
ρ182-22-22-202002-2-2-2020000000000    orthogonal lifted from D4
ρ192-22-22-20-200-2-222020000000000    orthogonal lifted from D4
ρ202222220200-2-22-20-20000000000    orthogonal lifted from D4
ρ214-44-4-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400202-20000-200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ2444-4-400-20-220000200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ254-4-440020-2-20000200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ264-4-4400-20220000-200000000000    symplectic lifted from C42.3C4, Schur index 2

Smallest permutation representation of C2×C42.3C4
►On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

(2 28 6 32)(4 26 8 30)(9 24 13 20)(11 22 15 18)

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

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

Matrix representation of C2×C42.3C4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
15160000
001000
000100
00001016
0000167
,
1600000
0160000
007100
0011000
00001016
0000167
,
110000
0160000
0000160
0000016
0001600
001000

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,16,7],[1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,16,0,0] >;
 

C2×C42.3C4 in GAP, Magma, Sage, TeX 

C_2\times C_4^2._3C_4
 
% in TeX
 
G:=Group("C2xC4^2.3C4");
 
// GroupNames label
 
G:=SmallGroup(128,863);
 
// by ID
 
G=gap.SmallGroup(128,863);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1018,248,1971,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
 
// generators/relations
 

Export

Character table of C2×C42.3C4 in TeX

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