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G = C4⋊SD16order 64 = 26

The semidirect product of C4 and SD16 acting via SD16/Q8=C2

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

Aliases: Q81D4, C43SD16, C42.17C22, C4⋊C88C2, (C4×Q8)⋊3C2, C4.30(C2×D4), (C2×C4).130D4, C41D4.3C2, D4⋊C410C2, C2.7(C2×SD16), (C2×SD16)⋊11C2, C4.40(C4○D4), C4⋊C4.57C22, (C2×C8).29C22, (C2×C4).88C23, C22.84(C2×D4), C2.12(C4⋊D4), C2.10(C8⋊C22), (C2×D4).12C22, (C2×Q8).50C22, SmallGroup(64,141)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4⋊SD16
C1C2C4C2×C4C2×Q8C4×Q8 — C4⋊SD16
C1C2C2×C4 — C4⋊SD16
C1C22C42 — C4⋊SD16
C1C2C2C2×C4 — C4⋊SD16

Generators and relations for C4⋊SD16
 G = < a,b,c | a4=b8=c2=1, bab-1=cac=a-1, cbc=b3 >

Subgroups: 133 in 64 conjugacy classes, 29 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], Q8, C23 [×2], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C4⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22, C4⋊SD16

Character table of C4⋊SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111882222444444444
ρ11111111111111111111    trivial
ρ21111-1-11111-1-1-1-111111    linear of order 2
ρ311111-1-11-11-1-111-1-11-11    linear of order 2
ρ41111-11-11-1111-1-1-1-11-11    linear of order 2
ρ511111-1-11-1111-1-1-11-11-1    linear of order 2
ρ61111-11-11-11-1-111-11-11-1    linear of order 2
ρ71111111111-1-1-1-11-1-1-1-1    linear of order 2
ρ81111-1-1111111111-1-1-1-1    linear of order 2
ρ9222200-2-2-2-2000020000    orthogonal lifted from D4
ρ102222002-22-20000-20000    orthogonal lifted from D4
ρ112-22-2000-20200-2200000    orthogonal lifted from D4
ρ122-22-2000-202002-200000    orthogonal lifted from D4
ρ132-22-200020-2-2i2i0000000    complex lifted from C4○D4
ρ142-22-200020-22i-2i0000000    complex lifted from C4○D4
ρ152-2-220020-2000000-2-2--2--2    complex lifted from SD16
ρ162-2-2200-202000000-2--2--2-2    complex lifted from SD16
ρ172-2-220020-2000000--2--2-2-2    complex lifted from SD16
ρ182-2-2200-202000000--2-2-2--2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C4⋊SD16
On 32 points
Generators in S32
(1 26 18 11)(2 12 19 27)(3 28 20 13)(4 14 21 29)(5 30 22 15)(6 16 23 31)(7 32 24 9)(8 10 17 25)
(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)
(2 4)(3 7)(6 8)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 23)(19 21)(20 24)

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

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

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

C4⋊SD16 is a maximal subgroup of
C42.443D4  C42.444D4  C42.223D4  C42.450D4  C42.230D4  C42.233D4  C42.270D4  C42.274D4  C42.294D4  C42.302D4
 Q8⋊D4p: Q8⋊D8  Q83D8  C813SD16  Q82D12  Q8⋊D12  Q8⋊D20  Q8⋊D28 ...
 C4p⋊SD16: C814SD16  C8⋊SD16  C82SD16  Dic68D4  Dic69D4  Dic108D4  Dic109D4  Dic148D4 ...
 (Cp×Q8)⋊D4: C42.212D4  C42.16C23  Dic35SD16  Dic55SD16  Dic75SD16 ...
 C8pD4⋊C2: D4⋊SD16  C42.266D4  C42.275D4  C42.408C23  C42.410C23  C42.295D4  C42.299D4  C4.2- 1+4 ...
 C4⋊C4.D2p: C42.181C23  Q8⋊SD16  Q86SD16  C42.189C23  Q8.2SD16  Q8.2D8  C42.249C23  C42.253C23 ...
C4⋊SD16 is a maximal quotient of
C42.99D4  C42.118D4  C4⋊C47D4  (C2×C4)⋊5SD16  (C2×C4).19Q16
 Q8⋊D4p: C813SD16  Q82D12  Q8⋊D20  Q8⋊D28 ...
 C4p⋊SD16: C814SD16  C8⋊SD16  C82SD16  Dic68D4  Dic69D4  Dic108D4  Dic109D4  Dic148D4 ...
 (Cp×Q8)⋊D4: (C2×C8)⋊20D4  Dic35SD16  Dic55SD16  Dic75SD16 ...
 C4⋊C4.D2p: Q81Q16  C8.SD16  (C2×SD16)⋊15C4  C4.67(C4×D4)  C42.30Q8  C4⋊C4.106D4  C2.(C83Q8)  Dic62D4 ...

Matrix representation of C4⋊SD16 in GL4(𝔽17) generated by

16000
01600
00138
0004
,
121200
51200
00115
00116
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,8,4],[12,5,0,0,12,12,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C4⋊SD16 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C4:SD16");
// GroupNames label

G:=SmallGroup(64,141);
// by ID

G=gap.SmallGroup(64,141);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,158,1444,376,88]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

Export

Character table of C4⋊SD16 in TeX

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