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G = C4×SD16order 64 = 26

Direct product of C4 and SD16

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

Aliases: C4×SD16, C42.72C22, C85(C2×C4), (C4×C8)⋊11C2, Q81(C2×C4), (C4×Q8)⋊1C2, C42(C4.Q8), C4.Q814C2, D4.1(C2×C4), (C4×D4).4C2, C2.13(C4×D4), (C2×C4).52D4, C2.4(C4○D8), C4.2(C4○D4), C43(D4⋊C4), C2.4(C2×SD16), C42(Q8⋊C4), Q8⋊C421C2, D4⋊C4.8C2, C4⋊C4.51C22, (C2×C8).63C22, C4.10(C22×C4), (C2×C4).74C23, (C2×SD16).5C2, C22.52(C2×D4), (C2×D4).51C22, (C2×Q8).44C22, SmallGroup(64,119)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×SD16
C1C2C22C2×C4C42C4×Q8 — C4×SD16
C1C2C4 — C4×SD16
C1C2×C4C42 — C4×SD16
C1C2C2C2×C4 — C4×SD16

Generators and relations for C4×SD16
 G = < a,b,c | a4=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 101 in 61 conjugacy classes, 37 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C4×SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, SD16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×SD16, C4○D8, C4×SD16

Character table of C4×SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 1111441111222244444422222222
ρ11111111111111111111111111111    trivial
ρ2111111-1-1-1-11-1-11-1-1-111-11-11-11-11-1    linear of order 2
ρ31111-1-1-1-1-1-11-1-11-11-1111-11-11-11-11    linear of order 2
ρ41111-1-1111111111-1111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-1-11-1-11111-1-111-11-11-11-1    linear of order 2
ρ61111-1-111111111-1-1-1-1-1-111111111    linear of order 2
ρ711111111111111-11-1-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111-1-1-1-11-1-111-11-1-1-1-11-11-11-11    linear of order 2
ρ91-1-11-11-iii-i1i-i-1-1-i1i-ii-i1-i1i-1i-1    linear of order 4
ρ101-1-111-1i-i-ii1-ii-11-i-1i-iii1i1-i-1-i-1    linear of order 4
ρ111-1-111-1i-i-ii1-ii-1-1-i1-iii-i-1-i-1i1i1    linear of order 4
ρ121-1-11-11-iii-i1i-i-11-i-1-iiii-1i-1-i1-i1    linear of order 4
ρ131-1-111-1-iii-i1i-i-1-1i1i-i-ii-1i-1-i1-i1    linear of order 4
ρ141-1-11-11i-i-ii1-ii-11i-1i-i-i-i-1-i-1i1i1    linear of order 4
ρ151-1-11-11i-i-ii1-ii-1-1i1-ii-ii1i1-i-1-i-1    linear of order 4
ρ161-1-111-1-iii-i1i-i-11i-1-ii-i-i1-i1i-1i-1    linear of order 4
ρ172222002222-2-2-2-200000000000000    orthogonal lifted from D4
ρ18222200-2-2-2-2-222-200000000000000    orthogonal lifted from D4
ρ192-2-22002i-2i-2i2i-22i-2i200000000000000    complex lifted from C4○D4
ρ202-2-2200-2i2i2i-2i-2-2i2i200000000000000    complex lifted from C4○D4
ρ212-22-200-22-220000000000--2--2-2-2--2--2-2-2    complex lifted from SD16
ρ2222-2-2002i2i-2i-2i0000000000-2--22-22-2-2--2    complex lifted from C4○D8
ρ232-22-2002-22-20000000000-2--2--2-2-2--2--2-2    complex lifted from SD16
ρ2422-2-200-2i-2i2i2i0000000000-2-22--22--2-2-2    complex lifted from C4○D8
ρ252-22-200-22-220000000000-2-2--2--2-2-2--2--2    complex lifted from SD16
ρ262-22-2002-22-20000000000--2-2-2--2--2-2-2--2    complex lifted from SD16
ρ2722-2-200-2i-2i2i2i00000000002--2-2-2-2-22--2    complex lifted from C4○D8
ρ2822-2-2002i2i-2i-2i00000000002-2-2--2-2--22-2    complex lifted from C4○D8

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

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

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

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

C4×SD16 is a maximal subgroup of
SD16⋊C8  C812SD16  C815SD16  D4.M4(2)  Q82M4(2)  C89SD16  C8⋊M4(2)  C42.222D4  C42.384D4  C42.223D4  C42.225D4  C42.450D4  C42.451D4  C42.226D4  C42.352C23  C42.353C23  C42.354C23  C42.355C23  C42.357C23  C42.359C23  C42.360C23  C42.308D4  C42.256D4  C42.385C23  C42.390C23  SD163D4  SD1610D4  D47SD16  C42.461C23  D48SD16  C42.467C23  C42.472C23  C42.473C23  C42.478C23  C42.480C23  D49SD16  C42.486C23  C42.489C23  C42.492C23  C42.494C23  C42.498C23  Q87SD16  C42.501C23  C42.502C23  Q88SD16  C42.505C23  C42.506C23  C42.509C23  C42.510C23  C42.512C23  C42.513C23  C42.514C23  C42.517C23  SD164Q8  SD16⋊Q8  SD162Q8  SD163Q8  C42.73C23  C42.531C23
 C2p.(C4×D4): C42.275C23  C42.276C23  C42.278C23  C42.281C23  Dic36SD16  Dic37SD16  Dic38SD16  Dic56SD16 ...
 C8pD4⋊C2: C42.365D4  C42.255D4  C42.386C23  C42.391C23  SD161D4  SD162D4  SD1611D4  C42.466C23 ...
C4×SD16 is a maximal quotient of
C812SD16  C815SD16  C89SD16  C4.Q89C4  C4.Q810C4  C87(C4⋊C4)
 C2p.(C4×D4): D4⋊(C4⋊C4)  Q8⋊C4⋊C4  (C2×SD16)⋊14C4  (C2×SD16)⋊15C4  C4.67(C4×D4)  C4.68(C4×D4)  C2.(C88D4)  C2.(C87D4) ...

Matrix representation of C4×SD16 in GL3(𝔽17) generated by

1300
0160
0016
,
100
0107
050
,
100
010
0116
G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,0,16],[1,0,0,0,10,5,0,7,0],[1,0,0,0,1,1,0,0,16] >;

C4×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("C4xSD16");
// GroupNames label

G:=SmallGroup(64,119);
// by ID

G=gap.SmallGroup(64,119);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,86,963,489,117]);
// Polycyclic

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

Export

Character table of C4×SD16 in TeX

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