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G = C4⋊2Q16  order 64 = 26

The semidirect product of C4 and Q16 acting via Q16/Q8=C2

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

Aliases: C4⋊2Q16, Q8.1D4, C42.19C22, C4⋊C8.6C2, C4⋊Q8.4C2, (C2×C4).27D4, C4.32(C2×D4), (C4×Q8).5C2, C2.5(C2×Q16), (C2×C8).4C22, (C2×Q16).2C2, C4.42(C4○D4), C4⋊C4.59C22, (C2×C4).90C23, Q8⋊C4.2C2, C22.86(C2×D4), (C2×Q8).8C22, C2.14(C4⋊D4), C2.10(C8.C22), SmallGroup(64,143)

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

Derived series
C1 — C2×C4 — C4⋊2Q16
Chief series
C1 — C2 — C4 — C2×C4 — C2×Q8 — C4×Q8 — C4⋊2Q16
Lower central
C1 — C2 — C2×C4 — C4⋊2Q16
Upper central
C1 — C22 — C42 — C4⋊2Q16
Jennings
C1 — C2 — C2 — C2×C4 — C4⋊2Q16

Generators and relations for C4⋊2Q16
 G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

C1
C2
C2
C2
C4
C22
C4
C4
C4
2C4
2C4
2C4
4C4
4C4
4C4
Q8
C2×C4
C2×C4
Q8
C2×C4
2Q8
2C2×C4
2Q8
2C8
2C2×C4
2C8
2Q8
2Q8
2C2×C4
2C2×C4
2Q8
C2×Q8
C2×C8
C2×Q8
C2×Q8
C2×C8
C4⋊C4
C42
2Q16
2Q16
2Q16
2Q16
2C4⋊C4
2C4⋊C4
2C4⋊C4
2C42
Q8⋊C4
Q8⋊C4
C2×Q16
C2×Q16
C4×Q8
C4⋊C8
C4⋊Q8
C4⋊2Q16

Character table of C4⋊2Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ211111111-11-1-1-1-1-11111    linear of order 2
ρ31111-111-1-1-1-1111-11-1-11    linear of order 2
ρ41111-111-11-11-1-1-111-1-11    linear of order 2
ρ51111-111-1-1-1-111-11-111-1    linear of order 2
ρ61111-111-11-11-1-11-1-111-1    linear of order 2
ρ71111111111111-1-1-1-1-1-1    linear of order 2
ρ811111111-11-1-1-111-1-1-1-1    linear of order 2
ρ92-22-202-200002-2000000    orthogonal lifted from D4
ρ1022222-2-220-2000000000    orthogonal lifted from D4
ρ112-22-202-20000-22000000    orthogonal lifted from D4
ρ122222-2-2-2-202000000000    orthogonal lifted from D4
ρ132-2-22200-20000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ142-2-22200-20000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ152-2-22-20020000000√2-√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-20020000000-√2√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ172-22-20-2202i0-2i00000000    complex lifted from C4○D4
ρ182-22-20-220-2i02i00000000    complex lifted from C4○D4
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C4⋊2Q16
►Regular action on 64 points
Generators in S64
(1 42 19 35)(2 36 20 43)(3 44 21 37)(4 38 22 45)(5 46 23 39)(6 40 24 47)(7 48 17 33)(8 34 18 41)(9 63 51 30)(10 31 52 64)(11 57 53 32)(12 25 54 58)(13 59 55 26)(14 27 56 60)(15 61 49 28)(16 29 50 62)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 60 45 64)(42 59 46 63)(43 58 47 62)(44 57 48 61)
 
G:=sub<Sym(64)| (1,42,19,35)(2,36,20,43)(3,44,21,37)(4,38,22,45)(5,46,23,39)(6,40,24,47)(7,48,17,33)(8,34,18,41)(9,63,51,30)(10,31,52,64)(11,57,53,32)(12,25,54,58)(13,59,55,26)(14,27,56,60)(15,61,49,28)(16,29,50,62), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,60,45,64)(42,59,46,63)(43,58,47,62)(44,57,48,61)>;
 
G:=Group( (1,42,19,35)(2,36,20,43)(3,44,21,37)(4,38,22,45)(5,46,23,39)(6,40,24,47)(7,48,17,33)(8,34,18,41)(9,63,51,30)(10,31,52,64)(11,57,53,32)(12,25,54,58)(13,59,55,26)(14,27,56,60)(15,61,49,28)(16,29,50,62), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,60,45,64)(42,59,46,63)(43,58,47,62)(44,57,48,61) );
 
G=PermutationGroup([[(1,42,19,35),(2,36,20,43),(3,44,21,37),(4,38,22,45),(5,46,23,39),(6,40,24,47),(7,48,17,33),(8,34,18,41),(9,63,51,30),(10,31,52,64),(11,57,53,32),(12,25,54,58),(13,59,55,26),(14,27,56,60),(15,61,49,28),(16,29,50,62)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(41,60,45,64),(42,59,46,63),(43,58,47,62),(44,57,48,61)]])
 

C4⋊2Q16 is a maximal subgroup of
 C42.443D4  C42.212D4  C42.445D4  C42.224D4  C42.451D4  C42.231D4  C42.234D4  C42.267D4  C42.270D4  C42.274D4  C42.276D4  C42.296D4  C42.297D4  C42.300D4  C42.303D4  Q8.D12
 C4p⋊Q16: C8⋊8Q16  C8⋊7Q16  C8⋊Q16  C8⋊2Q16  C4⋊Dic12  C12⋊7Q16  C12⋊Q16  C4⋊Dic20 ...
 (Cp×Q8).D4: C42.201C23  Q8.D8  D4⋊3Q16  Q8⋊3Q16  Q8⋊4Q16  D4⋊4Q16  C42.213C23  Q8.SD16 ...
 C4⋊C4.D2p: C42.19C23  C42.354C23  C42.361C23  C42.409C23  C42.411C23  C42.25C23  C42.28C23  SD16⋊8D4 ...
C4⋊2Q16 is a maximal quotient of
 C2.(C4×Q16)  C42.29Q8  C42.117D4  (C2×C8).1Q8
 C4p⋊Q16: C8⋊8Q16  C8⋊7Q16  C8⋊Q16  C8⋊2Q16  C4⋊Dic12  C12⋊7Q16  C12⋊Q16  C4⋊Dic20 ...
 (Cp×Q8).D4: Q8.1Q16  C8.3Q16  C42.99D4  (C2×C4)⋊9Q16  (C2×C4)⋊2Q16  C4⋊C4.95D4  (C2×C4)⋊3Q16  (C2×C4).19Q16 ...
 (C2×C8).D2p: (C2×C8).52D4  Dic3⋊Q16  Dic5⋊Q16  Dic7⋊Q16 ...

Matrix representation of C4⋊2Q16 ►in GL4(𝔽17) generated by

16000
01600
0019
001316
,
31400
3300
00130
00164
,
4000
01300
0042
00113
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,13,0,0,9,16],[3,3,0,0,14,3,0,0,0,0,13,16,0,0,0,4],[4,0,0,0,0,13,0,0,0,0,4,1,0,0,2,13] >;
 

C4⋊2Q16 in GAP, Magma, Sage, TeX 

C_4\rtimes_2Q_{16}
 
% in TeX
 
G:=Group("C4:2Q16");
 
// GroupNames label
 
G:=SmallGroup(64,143);
 
// by ID
 
G=gap.SmallGroup(64,143);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,158,1444,376,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4⋊2Q16 in TeX
Character table of C4⋊2Q16 in TeX

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