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G = Q32⋊C4order 128 = 27

The semidirect product of Q32 and C4 acting faithfully

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

Aliases: Q32⋊C4, SD321C4, C42.11D4, M5(2).12C22, C16.(C2×C4), D8.(C2×C4), Q16.(C2×C4), C8.Q81C2, C8.26D4.C2, C16⋊C41C2, C4.33(C4×D4), (C2×C8).31D4, (C2×C8).1C23, Q16⋊C41C2, D82C4.3C2, C8.25(C4○D4), C8.17D46C2, 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

C1C8 — Q32⋊C4
C1C2C4C2×C4C2×C8C8⋊C4C8.26D4 — Q32⋊C4
C1C2C4C8 — Q32⋊C4
C1C2C2×C4C8⋊C4 — Q32⋊C4
C1C2C2C2C2C4C4C2×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 [×2], C4 [×2], C4 [×5], C22, C22, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×4], C16 [×2], C16, C42, C42, C4⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], D8, SD16, Q16, Q16 [×2], Q16, C2×Q8, C4○D4, C8⋊C4, Q8⋊C4, C4≀C2, C4.Q8, C8.C4, M5(2) [×2], SD32 [×2], Q32 [×2], C4×Q8, C8○D4, C2×Q16, C4○D8, C16⋊C4, D82C4, C8.17D4, C8.Q8, Q16⋊C4, C8.26D4, Q32⋊C2, Q32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22 [×2], 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 31 9 23)(2 30 10 22)(3 29 11 21)(4 28 12 20)(5 27 13 19)(6 26 14 18)(7 25 15 17)(8 24 16 32)
(1 20)(2 17 10 25)(3 30)(4 27 12 19)(5 24)(6 21 14 29)(7 18)(8 31 16 23)(9 28)(11 22)(13 32)(15 26)

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

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

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

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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