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G = (C2×Q8).Q8  order 128 = 27

2nd non-split extension by C2×Q8 of Q8 acting via Q8/C2=C22

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

Aliases: (C2×Q8).2Q8, (C2×C4).2C42, (C2×Q8).40D4, C4.10D4⋊5C4, (C2×C42).13C4, (C22×C4).38D4, (C22×Q8).1C22, C22.15(C23⋊C4), C2.3(C42.C4), C2.3(C42.3C4), C23.162(C22⋊C4), C2.16(C23.9D4), C23.67C23.3C2, C22.5(C2.C42), (C2×C4⋊C4).10C4, (C2×C4).3(C4⋊C4), (C2×Q8).41(C2×C4), (C2×C4).3(C22⋊C4), (C22×C4).67(C2×C4), (C2×C4.10D4).5C2, SmallGroup(128,126)

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

Derived series
C1 — C2×C4 — (C2×Q8).Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C23.67C23 — (C2×Q8).Q8
Lower central
C1 — C2 — C22 — C2×C4 — (C2×Q8).Q8
Upper central
C1 — C22 — C23 — C22×Q8 — (C2×Q8).Q8
Jennings
C1 — C2 — C22 — C22×Q8 — (C2×Q8).Q8

Generators and relations for (C2×Q8).Q8
 G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=ab-1d2, dbd-1=ab=ba, ece-1=ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, be=eb, cd=dc, ede-1=cd-1 >

Subgroups: 192 in 86 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, C42, C4⋊C4, C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×2], C2.C42 [×2], C4.10D4 [×4], C4.10D4 [×2], C2×C42, C2×C4⋊C4, C2×M4(2) [×2], C22×Q8, C23.67C23, C2×C4.10D4 [×2], (C2×Q8).Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, C42.C4, C42.3C4, (C2×Q8).Q8

Character table of (C2×Q8).Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111-11111-11-1-11-1-1-1-11111-1-1    linear of order 2
ρ3111111-11111-11-1-11-1-111-1-1-1-111    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1-11i1-1-11i-1-i-i1i-i11-i-iii-1-1    linear of order 4
ρ6111111-111-1-1-1-1-1-1-111-iii-ii-i-ii    linear of order 4
ρ7111111111-1-11-111-1-1-1i-ii-ii-ii-i    linear of order 4
ρ81-11-1-11-i1-11-1-i1ii-1i-i-ii1-1-11i-i    linear of order 4
ρ91-11-1-11i1-1-11i-1-i-i1i-i-1-1ii-i-i11    linear of order 4
ρ10111111-111-1-1-1-1-1-1-111i-i-ii-iii-i    linear of order 4
ρ11111111111-1-11-111-1-1-1-ii-ii-ii-ii    linear of order 4
ρ121-11-1-11-i1-11-1-i1ii-1i-ii-i-111-1-ii    linear of order 4
ρ131-11-1-11-i1-1-11-i-1ii1-ii-1-1-i-iii11    linear of order 4
ρ141-11-1-11i1-11-1i1-i-i-1-iii-i1-1-11-ii    linear of order 4
ρ151-11-1-11-i1-1-11-i-1ii1-ii11ii-i-i-1-1    linear of order 4
ρ161-11-1-11i1-11-1i1-i-i-1-ii-ii-111-1i-i    linear of order 4
ρ172222220-2-2220-200-20000000000    orthogonal lifted from D4
ρ182222220-2-2-2-2020020000000000    orthogonal lifted from D4
ρ192-22-2-220-22-220200-20000000000    orthogonal lifted from D4
ρ202-22-2-220-222-20-20020000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-44-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400-20000202-200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ2444-4-40020000-20-2200000000000    symplectic lifted from C42.3C4, Schur index 2
ρ254-4-44002i0000-2i02i-2i00000000000    complex lifted from C42.C4
ρ264-4-4400-2i00002i0-2i2i00000000000    complex lifted from C42.C4

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

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

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

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

Matrix representation of (C2×Q8).Q8 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
004000
0001300
000040
0000013
,
100000
010000
0001600
001000
000001
0000160
,
400000
0130000
001000
000100
000001
0000160
,
0160000
1600000
000010
000001
0013000
000400

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

(C2×Q8).Q8 in GAP, Magma, Sage, TeX 

(C_2\times Q_8).Q_8
 
% in TeX
 
G:=Group("(C2xQ8).Q8");
 
// GroupNames label
 
G:=SmallGroup(128,126);
 
// by ID
 
G=gap.SmallGroup(128,126);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,248,2804,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=d^4=1,c^2=b^2,e^2=a*b^-1*d^2,d*b*d^-1=a*b=b*a,e*c*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=b^-1,b*e=e*b,c*d=d*c,e*d*e^-1=c*d^-1>;
 
// generators/relations
 

Export

Character table of (C2×Q8).Q8 in TeX

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