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G = C4⋊Q8.C4  order 128 = 27

5th non-split extension by C4⋊Q8 of C4 acting faithfully

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

Aliases: C4⋊Q8.5C4, C42.12(C2×C4), (C2×Q8).119D4, C4.4D4.1C4, C4⋊Q8.95C22, C42.3C4⋊6C2, C42.C4⋊6C2, (C22×C4).100D4, (C2×Q8).13C23, C42⋊C2.10C4, C23.85(C22⋊C4), C22.10(C23⋊C4), C4.10D4.7C22, C4.4D4.16C22, (C22×Q8).86C22, C23.38C23.8C2, (C2×C4).10(C2×D4), (C2×C4○D4).10C4, (C2×D4).41(C2×C4), C2.45(C2×C23⋊C4), (C22×C4).34(C2×C4), (C2×Q8).109(C2×C4), (C2×C4.10D4)⋊27C2, (C2×C4).31(C22⋊C4), (C2×C4).102(C22×C4), C22.69(C2×C22⋊C4), SmallGroup(128,865)

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

Derived series
C1 — C2×C4 — C4⋊Q8.C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C23.38C23 — C4⋊Q8.C4
Lower central
C1 — C2 — C22 — C2×C4 — C4⋊Q8.C4
Upper central
C1 — C2 — C23 — C22×Q8 — C4⋊Q8.C4
Jennings
C1 — C2 — C22 — C2×Q8 — C4⋊Q8.C4

Generators and relations for C4⋊Q8.C4
 G = < a,b,c,d | a4=b4=1, c2=d4=b2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b, cd=dc >

Subgroups: 252 in 115 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C22, 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, C4.10D4, C4.10D4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C42.C4, C42.3C4, C2×C4.10D4, C23.38C23, C4⋊Q8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4⋊Q8.C4

Character table of C4⋊Q8.C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11222844444488888888888
ρ111111111111111111111111    trivial
ρ211-11-1-1-1111-1-11-11-11-1-1111-1    linear of order 2
ρ311-11-11-1111-1-1-11-111-11-1-11-1    linear of order 2
ρ411111-1111111-1-1-1-111-1-1-111    linear of order 2
ρ511-11-11-1111-1-1-11-1-1-11-111-11    linear of order 2
ρ611111-1111111-1-1-11-1-1111-1-1    linear of order 2
ρ7111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811-11-1-1-1111-1-11-111-111-1-1-11    linear of order 2
ρ911-11-111-11-11-1-1-11i-ii-ii-ii-i    linear of order 4
ρ1011-11-1-11-11-11-111-1-i-iii-iii-i    linear of order 4
ρ11111111-1-11-1-111-1-1i-i-i-i-iiii    linear of order 4
ρ1211111-1-1-11-1-11-111-i-i-iii-iii    linear of order 4
ρ1311-11-111-11-11-1-1-11-ii-ii-ii-ii    linear of order 4
ρ1411-11-1-11-11-11-111-1ii-i-ii-i-ii    linear of order 4
ρ15111111-1-11-1-111-1-1-iiiii-i-i-i    linear of order 4
ρ1611111-1-1-11-1-11-111iii-i-ii-i-i    linear of order 4
ρ1722-22-2022-2-2-2200000000000    orthogonal lifted from D4
ρ182222202-2-22-2-200000000000    orthogonal lifted from D4
ρ19222220-22-2-22-200000000000    orthogonal lifted from D4
ρ2022-22-20-2-2-222200000000000    orthogonal lifted from D4
ρ21444-4-4000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4-44000000000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C4⋊Q8.C4
►On 32 points
Generators in S32
(1 25)(2 8 6 4)(3 31)(5 29)(7 27)(9 11 13 15)(10 23)(12 21)(14 19)(16 17)(18 24 22 20)(26 28 30 32)

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

(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 31 21 27)(18 32 22 28)(19 25 23 29)(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)| (1,25)(2,8,6,4)(3,31)(5,29)(7,27)(9,11,13,15)(10,23)(12,21)(14,19)(16,17)(18,24,22,20)(26,28,30,32), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,17,14,21)(11,18,15,22)(12,23,16,19), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,31,21,27)(18,32,22,28)(19,25,23,29)(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( (1,25)(2,8,6,4)(3,31)(5,29)(7,27)(9,11,13,15)(10,23)(12,21)(14,19)(16,17)(18,24,22,20)(26,28,30,32), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,17,14,21)(11,18,15,22)(12,23,16,19), (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9)(17,31,21,27)(18,32,22,28)(19,25,23,29)(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([[(1,25),(2,8,6,4),(3,31),(5,29),(7,27),(9,11,13,15),(10,23),(12,21),(14,19),(16,17),(18,24,22,20),(26,28,30,32)], [(1,31,5,27),(2,32,6,28),(3,29,7,25),(4,30,8,26),(9,20,13,24),(10,17,14,21),(11,18,15,22),(12,23,16,19)], [(1,10,5,14),(2,11,6,15),(3,12,7,16),(4,13,8,9),(17,31,21,27),(18,32,22,28),(19,25,23,29),(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 C4⋊Q8.C4 ►in GL8(𝔽17)

000160000
00100000
01000000
160000000
000001600
00001000
161611161162
01610160161
,
00100000
00010000
160000000
016000000
000000160
161611161162
00001000
1100116016
,
000130000
001300000
013000000
130000000
131344134138
000000130
000001300
00000004
,
00001000
00000100
00000010
111616116115
01000000
160000000
000160000
000001161

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

C4⋊Q8.C4 in GAP, Magma, Sage, TeX 

C_4\rtimes Q_8.C_4
 
% in TeX
 
G:=Group("C4:Q8.C4");
 
// GroupNames label
 
G:=SmallGroup(128,865);
 
// by ID
 
G=gap.SmallGroup(128,865);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^2=d^4=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,c*d=d*c>;
 
// generators/relations
 

Export

Character table of C4⋊Q8.C4 in TeX

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