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G = (C2×D4).137D4  order 128 = 27

99th non-split extension by C2×D4 of D4 acting via D4/C2=C22

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

Aliases: (C2×D4).137D4, C42.14(C2×C4), C4.4D4.2C4, C4⋊Q8.96C22, C4.21(C23⋊C4), C42.3C4⋊7C2, (C22×C4).102D4, (C2×Q8).15C23, (C22×Q8).14C4, C42⋊C2.12C4, C23.27(C22⋊C4), C4.10D4.8C22, C23.38C23.9C2, M4(2).8C22.11C2, (C2×C4).12(C2×D4), C2.47(C2×C23⋊C4), (C2×Q8).40(C2×C4), (C2×D4).128(C2×C4), (C22×C4).36(C2×C4), (C2×C4).104(C22×C4), (C2×C4○D4).79C22, C22.71(C2×C22⋊C4), (C2×C4).147(C22⋊C4), SmallGroup(128,867)

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

Derived series
C1 — C2×C4 — (C2×D4).137D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C2×C4○D4 — C23.38C23 — (C2×D4).137D4
Lower central
C1 — C2 — C22 — C2×C4 — (C2×D4).137D4
Upper central
C1 — C2 — C2×C4 — C2×C4○D4 — (C2×D4).137D4
Jennings
C1 — C2 — C22 — C2×Q8 — (C2×D4).137D4

Generators and relations for (C2×D4).137D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=ab-1, dbd-1=ab=ba, dcd-1=ece-1=ac=ca, dad-1=eae-1=ab2, cbc=ebe-1=b-1, ede-1=ab-1d3 >

Subgroups: 252 in 115 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.D4, C4.10D4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C42.3C4, M4(2).8C22, C23.38C23, (C2×D4).137D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, (C2×D4).137D4

Character table of (C2×D4).137D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11244422444888888888888
ρ111111111111111111111111    trivial
ρ2111-1-1-1-1-111111-1-1-11-11-11-11    linear of order 2
ρ311111111111-1-1-1-1111-1-1-1-11    linear of order 2
ρ4111-1-1-1-1-1111-1-111-11-1-11-111    linear of order 2
ρ5111-1-1-1-1-1111-1-1111-111-11-1-1    linear of order 2
ρ611111111111-1-1-1-1-1-1-11111-1    linear of order 2
ρ7111-1-1-1-1-111111-1-11-11-11-11-1    linear of order 2
ρ8111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91111-11-1-1-11-1-11-11-i-iii-i-iii    linear of order 4
ρ10111-11-111-11-11-1-11i-i-i-i-iiii    linear of order 4
ρ11111-11-111-11-1-111-1i-i-iii-i-ii    linear of order 4
ρ121111-11-1-1-11-11-11-1-i-ii-iii-ii    linear of order 4
ρ131111-11-1-1-11-1-11-11ii-i-iii-i-i    linear of order 4
ρ14111-11-111-11-11-1-11-iiiii-i-i-i    linear of order 4
ρ15111-11-111-11-1-111-1-iii-i-iii-i    linear of order 4
ρ161111-11-1-1-11-11-11-1ii-ii-i-ii-i    linear of order 4
ρ172222-2-2222-2-2000000000000    orthogonal lifted from D4
ρ18222-2-2222-2-22000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-22000000000000    orthogonal lifted from D4
ρ20222-222-2-22-2-2000000000000    orthogonal lifted from D4
ρ2144-40004-4000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-44000000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×D4).137D4
►On 32 points
Generators in S32
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)

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

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

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

Matrix representation of (C2×D4).137D4 ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
13161400100
1312100010
041300001
,
01000000
160000000
1616120000
0116160000
131614016200
1300316100
134131616101
000401160
,
1616120000
00100000
01000000
016110000
0413016002
13161416011
13161416101
041300001
,
000013000
1641204900
1631304090
01160013413
1313480000
13134401350
13130401440
13134401610
,
00001000
413011500
4516010150
000001611
01000000
0000416140
01161641210
000004130

G:=sub<GL(8,GF(17))| [16,0,0,0,0,13,13,0,0,16,0,0,0,16,12,4,0,0,16,0,0,14,1,13,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,16,0,13,13,13,0,1,0,16,1,16,0,4,0,0,0,1,16,14,0,13,0,0,0,2,16,0,3,16,4,0,0,0,0,16,16,16,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[16,0,0,0,0,13,13,0,16,0,1,16,4,16,16,4,1,1,0,1,13,1,1,13,2,0,0,1,0,4,4,0,0,0,0,0,16,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,1,1],[0,16,16,0,13,13,13,13,0,4,3,1,13,13,13,13,0,12,13,16,4,4,0,4,0,0,0,0,8,4,4,4,13,4,4,0,0,0,0,0,0,9,0,13,0,13,14,16,0,0,9,4,0,5,4,1,0,0,0,13,0,0,0,0],[0,4,4,0,0,0,0,0,0,1,5,0,1,0,1,0,0,3,16,0,0,0,16,0,0,0,0,0,0,0,16,0,1,1,1,0,0,4,4,0,0,15,0,16,0,16,12,4,0,0,15,1,0,14,1,13,0,0,0,1,0,0,0,0] >;
 

(C2×D4).137D4 in GAP, Magma, Sage, TeX 

(C_2\times D_4)._{137}D_4
 
% in TeX
 
G:=Group("(C2xD4).137D4");
 
// GroupNames label
 
G:=SmallGroup(128,867);
 
// by ID
 
G=gap.SmallGroup(128,867);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,1018,248,1971,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=a*b^-1,d*b*d^-1=a*b=b*a,d*c*d^-1=e*c*e^-1=a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,e*d*e^-1=a*b^-1*d^3>;
 
// generators/relations
 

Export

Character table of (C2×D4).137D4 in TeX

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