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G = Q32⋊C4  order 128 = 27

The semidirect product of Q32 and C4 acting faithfully

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

Aliases: Q32⋊C4, SD32⋊1C4, C42.11D4, M5(2).12C22, C16.(C2×C4), D8.(C2×C4), Q16.(C2×C4), C8.Q8⋊1C2, C8.26D4.C2, C16⋊C4⋊1C2, C4.33(C4×D4), (C2×C8).31D4, (C2×C8).1C23, Q16⋊C4⋊1C2, D8⋊2C4.3C2, C8.25(C4○D4), C8.17D4⋊6C2, C8.11(C22×C4), Q32⋊C2.1C2, C4○D8.1C22, C4.95(C8⋊C22), C8⋊C4.3C22, C4.Q8.1C22, C2.13(D8⋊C4), C8.C4.1C22, (C2×Q16).43C22, C22.22(C8⋊C22), (C2×C4).276(C2×D4), 2-Sylow(ASigmaL(2,81)), SmallGroup(128,912)

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

Derived series
C1 — C8 — Q32⋊C4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C8.26D4 — Q32⋊C4
Lower central
C1 — C2 — C4 — C8 — Q32⋊C4
Upper central
C1 — C2 — C2×C4 — C8⋊C4 — Q32⋊C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q32⋊C4

Generators and relations for Q32⋊C4
 G = < a,b,c | a16=c4=1, b2=a8, bab-1=a-1, cac-1=a5, cbc-1=a12b >

Subgroups: 148 in 71 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C16, C16, C42, C42, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C8⋊C4, Q8⋊C4, C4≀C2, C4.Q8, C8.C4, M5(2), SD32, Q32, C4×Q8, C8○D4, C2×Q16, C4○D8, C16⋊C4, D8⋊2C4, C8.17D4, C8.Q8, Q16⋊C4, C8.26D4, Q32⋊C2, Q32⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22, D8⋊C4, Q32⋊C4

Character table of Q32⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D
 size 11282244888884444888888
ρ111111111111111111111111    trivial
ρ211111111-1-1-11-1111111-1-1-1-1    linear of order 2
ρ3111-11111-1-1-1-1-11111-1-11111    linear of order 2
ρ4111-11111111-111111-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-11-1111-11-1-1-1-11-11    linear of order 2
ρ6111111-1-11-111-11-11-1-1-11-11-1    linear of order 2
ρ7111-111-1-11-11-1-11-11-111-11-11    linear of order 2
ρ8111-111-1-1-11-1-111-11-1111-11-1    linear of order 2
ρ911-11-11i-i-1-i1-1i-1-i1i-ii-1-i1i    linear of order 4
ρ1011-11-11-ii1-i-1-1i-1i1-ii-i1-i-1i    linear of order 4
ρ1111-11-11i-i1i-1-1-i-1-i1i-ii1i-1-i    linear of order 4
ρ1211-11-11-ii-1i1-1-i-1i1-ii-i-1i1-i    linear of order 4
ρ1311-1-1-11i-i1i-11-i-1-i1ii-i-1-i1i    linear of order 4
ρ1411-1-1-11-ii-1i11-i-1i1-i-ii1-i-1i    linear of order 4
ρ1511-1-1-11i-i-1-i11i-1-i1ii-i1i-1-i    linear of order 4
ρ1611-1-1-11-ii1-i-11i-1i1-i-ii-1i1-i    linear of order 4
ρ172220222200000-2-2-2-2000000    orthogonal lifted from D4
ρ18222022-2-200000-22-22000000    orthogonal lifted from D4
ρ1922-20-22-2i2i000002-2i-22i000000    complex lifted from C4○D4
ρ2022-20-222i-2i0000022i-2-2i000000    complex lifted from C4○D4
ρ2144-404-400000000000000000    orthogonal lifted from C8⋊C22
ρ224440-4-400000000000000000    orthogonal lifted from C8⋊C22
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of Q32⋊C4
►On 32 points
Generators in S32
(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 25 9 17)(2 24 10 32)(3 23 11 31)(4 22 12 30)(5 21 13 29)(6 20 14 28)(7 19 15 27)(8 18 16 26)

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

Matrix representation of Q32⋊C4 ►in GL8(𝔽17)

0000101313
0000016134
00001313016
0000134160
44010000
413100000
014130000
1013130000
,
00001000
00000100
00000010
00000001
160000000
016000000
001600000
000160000
,
014130000
160440000
13130160000
413100000
0000101313
000001413
0000134160
00001313016

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

Q32⋊C4 in GAP, Magma, Sage, TeX 

Q_{32}\rtimes C_4
 
% in TeX
 
G:=Group("Q32:C4");
 
// GroupNames label
 
G:=SmallGroup(128,912);
 
// by ID
 
G=gap.SmallGroup(128,912);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,268,1123,1466,521,136,1411,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=c^4=1,b^2=a^8,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^12*b>;
 
// generators/relations
 

Export

Character table of Q32⋊C4 in TeX

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