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G = C22×SD16order 64 = 26

Direct product of C22 and SD16

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

Aliases: C22×SD16, C83C23, C4.2C24, Q81C23, D4.1C23, C23.61D4, C4.17(C2×D4), (C2×C4).88D4, (C2×C8)⋊14C22, (C22×C8)⋊10C2, (C22×Q8)⋊8C2, (C2×Q8)⋊13C22, C22.65(C2×D4), C2.24(C22×D4), (C2×C4).136C23, (C2×D4).72C22, (C22×D4).12C2, (C22×C4).130C22, SmallGroup(64,251)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22×SD16
C1C2C4C2×C4C22×C4C22×D4 — C22×SD16
C1C2C4 — C22×SD16
C1C23C22×C4 — C22×SD16
C1C2C2C4 — C22×SD16

Generators and relations for C22×SD16
 G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 249 in 149 conjugacy classes, 89 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C22×SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C22×D4, C22×SD16

Character table of C22×SD16

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111111144442222444422222222
ρ11111111111111111111111111111    trivial
ρ21-1-1-111-11-111-1-11-111-11-1-11-1-111-11    linear of order 2
ρ311111111-1-1-1-11111-1-1-1-111111111    linear of order 2
ρ41-1-1-111-111-1-11-11-11-11-11-11-1-111-11    linear of order 2
ρ51-11-11-11-11-11-11-1-111-1-11-1-11-11-111    linear of order 2
ρ611-111-1-1-111-1-1-1-111-1-1111-1-111-1-11    linear of order 2
ρ711-111-1-1-1-1-111-1-11111-1-11-1-111-1-11    linear of order 2
ρ81-11-11-11-1-11-111-1-11-111-1-1-11-11-111    linear of order 2
ρ911111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ101-1-1-111-11-111-1-11-11-11-111-111-1-11-1    linear of order 2
ρ111-1-1-111-111-1-11-11-111-11-11-111-1-11-1    linear of order 2
ρ121111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ1311-111-1-1-111-1-1-1-11111-1-1-111-1-111-1    linear of order 2
ρ141-11-11-11-11-11-11-1-11-111-111-11-11-1-1    linear of order 2
ρ151-11-11-11-1-11-111-1-111-1-1111-11-11-1-1    linear of order 2
ρ1611-111-1-1-1-1-111-1-111-1-111-111-1-111-1    linear of order 2
ρ172-2-2-222-2200002-22-2000000000000    orthogonal lifted from D4
ρ1822-222-2-2-2000022-2-2000000000000    orthogonal lifted from D4
ρ192-22-22-22-20000-222-2000000000000    orthogonal lifted from D4
ρ20222222220000-2-2-2-2000000000000    orthogonal lifted from D4
ρ21222-2-22-2-2000000000000--2-2--2-2-2--2-2--2    complex lifted from SD16
ρ222-2-22-222-2000000000000--2--2--2-2--2-2-2-2    complex lifted from SD16
ρ2322-2-2-2-222000000000000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ242-222-2-2-22000000000000-2--2--2--2-2-2-2--2    complex lifted from SD16
ρ25222-2-22-2-2000000000000-2--2-2--2--2-2--2-2    complex lifted from SD16
ρ262-2-22-222-2000000000000-2-2-2--2-2--2--2--2    complex lifted from SD16
ρ2722-2-2-2-222000000000000--2--2-2-2-2-2--2--2    complex lifted from SD16
ρ282-222-2-2-22000000000000--2-2-2-2--2--2--2-2    complex lifted from SD16

Smallest permutation representation of C22×SD16
On 32 points
Generators in S32
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 28)(2 31)(3 26)(4 29)(5 32)(6 27)(7 30)(8 25)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)

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

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

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

C22×SD16 is a maximal subgroup of
(C2×SD16)⋊14C4  (C2×SD16)⋊15C4  (C2×C4)⋊9SD16  C8⋊(C22⋊C4)  M4(2).31D4  C233SD16  (C22×D8).C2  (C2×C4)⋊3SD16  (C2×C8)⋊20D4  (C2×C8).41D4  M4(2).5D4  C4⋊C4.97D4  (C2×C4)⋊5SD16  C42.278C23  D4.(C2×D4)  (C2×Q8)⋊16D4  C42.15C23  C42.16C23  (C2×C8)⋊11D4  M4(2)⋊10D4  SD16⋊D4  SD166D4  SD1610D4
C22×SD16 is a maximal quotient of
C42.222D4  C42.223D4  C42.365D4  C234SD16  C42.264D4  C42.266D4  C42.279D4  C42.281D4  C42.294D4  D47SD16  D48SD16  D49SD16  Q87SD16  Q88SD16  Q89SD16

Matrix representation of C22×SD16 in GL4(𝔽17) generated by

16000
0100
0010
0001
,
1000
01600
0010
0001
,
16000
0100
00125
001212
,
16000
0100
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,12,12,0,0,5,12],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16] >;

C22×SD16 in GAP, Magma, Sage, TeX

C_2^2\times {\rm SD}_{16}
% in TeX

G:=Group("C2^2xSD16");
// GroupNames label

G:=SmallGroup(64,251);
// by ID

G=gap.SmallGroup(64,251);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,192,217,1444,730,88]);
// Polycyclic

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

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Character table of C22×SD16 in TeX

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