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G = (C2×Q8).211D4order 128 = 27

19th non-split extension by C2×Q8 of D4 acting via D4/C22=C2

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

Aliases: (C2×Q8).211D4, C42⋊C2.9C4, (C22×C4).28C23, C23.66(C22×C4), (C22×Q8).9C22, M4(2)⋊4C4.2C2, C42⋊C2.6C22, C4.3(C22.D4), C22.8(C42⋊C2), C2.18(C23.34D4), (C2×M4(2)).165C22, C23.32C23.1C2, (C2×C4).236(C2×D4), C22⋊C4.51(C2×C4), (C22×C4).18(C2×C4), (C2×C4).313(C4○D4), (C2×C4).48(C22⋊C4), (C2×C4.10D4).7C2, C22.38(C2×C22⋊C4), SmallGroup(128,562)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — (C2×Q8).211D4
C1C2C4C2×C4C22×C4C42⋊C2C23.32C23 — (C2×Q8).211D4
C1C2C23 — (C2×Q8).211D4
C1C2C22×C4 — (C2×Q8).211D4
C1C2C2C22×C4 — (C2×Q8).211D4

Generators and relations for (C2×Q8).211D4
 G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=ab2c, ab=ba, ac=ca, dad-1=eae-1=ab2, cbc-1=b-1, dbd-1=ebe-1=ab-1, dcd-1=ece-1=b2c, ede-1=b2cd3 >

Subgroups: 204 in 118 conjugacy classes, 50 normal (8 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×8], C22, C22 [×2], C22, C8 [×4], C2×C4 [×10], C2×C4 [×8], Q8 [×8], C23, C42 [×6], C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C42⋊C2 [×6], C4×Q8 [×4], C2×M4(2) [×4], C22×Q8, M4(2)⋊4C4 [×4], C2×C4.10D4 [×2], C23.32C23, (C2×Q8).211D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, (C2×Q8).211D4

Character table of (C2×Q8).211D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F8G8H
 size 11222222244444444444488888888
ρ111111111111111111111111111111    trivial
ρ2111111111-1-1111-1-1-1-1-1-11-111-1-111-1    linear of order 2
ρ3111111111-11-1-1-1-1-1-1111-1-1-111-1-111    linear of order 2
ρ41111111111-1-1-1-1111-1-1-1-11-11-11-11-1    linear of order 2
ρ5111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111111-1-1111-1-1-1-1-1-111-1-111-1-11    linear of order 2
ρ7111111111-11-1-1-1-1-1-1111-111-1-111-1-1    linear of order 2
ρ81111111111-1-1-1-1111-1-1-1-1-11-11-11-11    linear of order 2
ρ911111-1-1-1-1-1-1-11-11-111-111-iiiii-i-i-i    linear of order 4
ρ1011111-1-1-1-111-11-1-11-1-11-11iii-i-i-i-ii    linear of order 4
ρ1111111-1-1-1-1-111-111-11-11-1-1-i-ii-iii-ii    linear of order 4
ρ1211111-1-1-1-11-11-11-11-11-11-1i-iii-ii-i-i    linear of order 4
ρ1311111-1-1-1-1-1-1-11-11-111-111i-i-i-i-iiii    linear of order 4
ρ1411111-1-1-1-111-11-1-11-1-11-11-i-i-iiiii-i    linear of order 4
ρ1511111-1-1-1-1-111-111-11-11-1-1ii-ii-i-ii-i    linear of order 4
ρ1611111-1-1-1-11-11-11-11-11-11-1-ii-i-ii-iii    linear of order 4
ρ1722-2-22-222-2200002-2-2000000000000    orthogonal lifted from D4
ρ1822-2-222-2-2220000-2-22000000000000    orthogonal lifted from D4
ρ1922-2-22-222-2-20000-222000000000000    orthogonal lifted from D4
ρ2022-2-222-2-22-2000022-2000000000000    orthogonal lifted from D4
ρ21222-2-22-22-20-2i0000002i2i-2i000000000    complex lifted from C4○D4
ρ2222-22-222-2-2002i-2i-2i0000002i00000000    complex lifted from C4○D4
ρ2322-22-2-2-22200-2i-2i2i0000002i00000000    complex lifted from C4○D4
ρ24222-2-22-22-202i000000-2i-2i2i000000000    complex lifted from C4○D4
ρ2522-22-2-2-222002i2i-2i000000-2i00000000    complex lifted from C4○D4
ρ26222-2-2-22-2202i0000002i-2i-2i000000000    complex lifted from C4○D4
ρ2722-22-222-2-200-2i2i2i000000-2i00000000    complex lifted from C4○D4
ρ28222-2-2-22-220-2i000000-2i2i2i000000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×Q8).211D4
On 32 points
Generators in S32
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 19 5 23)(2 24 6 20)(3 17 7 21)(4 22 8 18)(9 27 13 31)(10 32 14 28)(11 25 15 29)(12 30 16 26)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)
(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 10 25 22 5 14 29 18)(2 21 26 13 6 17 30 9)(3 16 27 20 7 12 31 24)(4 19 28 11 8 23 32 15)

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

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

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

Matrix representation of (C2×Q8).211D4 in GL8(𝔽17)

10000000
01000000
00100000
00010000
000016000
1340001600
000000160
1340000016
,
75570000
05500000
051200000
5012100000
1430012057
3003125012
1430057120
3003012125
,
016000000
10000000
1161620000
101610000
1340011500
040011600
1340000115
040000116
,
1430012057
314001210120
31400121050
6110012101210
05570000
12551203143
0121200000
7501203143
,
00001000
1340011500
00000010
000001601
01000000
100001300
1611150000
0011601300

G:=sub<GL(8,GF(17))| [1,0,0,0,0,13,0,13,0,1,0,0,0,4,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[7,0,0,5,14,3,14,3,5,5,5,0,3,0,3,0,5,5,12,12,0,0,0,0,7,0,0,10,0,3,0,3,0,0,0,0,12,12,5,0,0,0,0,0,0,5,7,12,0,0,0,0,5,0,12,12,0,0,0,0,7,12,0,5],[0,1,1,1,13,0,13,0,16,0,16,0,4,4,4,4,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[14,3,3,6,0,12,0,7,3,14,14,11,5,5,12,5,0,0,0,0,5,5,12,0,0,0,0,0,7,12,0,12,12,12,12,12,0,0,0,0,0,10,10,10,0,3,0,3,5,12,5,12,0,14,0,14,7,0,0,10,0,3,0,3],[0,13,0,0,0,1,16,0,0,4,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,15,0,16,0,13,0,13,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

(C2×Q8).211D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{211}D_4
% in TeX

G:=Group("(C2xQ8).211D4");
// GroupNames label

G:=SmallGroup(128,562);
// by ID

G=gap.SmallGroup(128,562);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2019,718,2028,124]);
// Polycyclic

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

Export

Character table of (C2×Q8).211D4 in TeX

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