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G = C4.6Q16  order 64 = 26

2nd non-split extension by C4 of Q16 acting via Q16/Q8=C2

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

Aliases: C4.6Q16, C4.7SD16, C42.5C22, C4⋊C8.3C2, C4⋊Q8.2C2, (C2×Q8).2C4, (C2×C4).110D4, C2.5(Q8⋊C4), C2.5(C4.D4), C22.41(C22⋊C4), (C2×C4).14(C2×C4), SmallGroup(64,14)

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

Derived series
C1 — C2×C4 — C4.6Q16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4⋊Q8 — C4.6Q16
Lower central
C1 — C22 — C2×C4 — C4.6Q16
Upper central
C1 — C22 — C42 — C4.6Q16
Jennings
C1 — C22 — C22 — C42 — C4.6Q16

Generators and relations for C4.6Q16
 G = < a,b,c | a4=b8=1, c2=a2b4, bab-1=cac-1=a-1, cbc-1=a-1b-1 >

C1
C2
C2
C2
C4
C4
C22
C4
C4
2C4
4C4
4C4
C2×C4
C2×C4
C2×C4
2C2×C4
2C2×C4
4Q8
4Q8
4C8
4C8
C2×Q8
C42
C2×Q8
2C2×C8
2C2×C8
2C4⋊C4
2C4⋊C4
C4⋊C8
C4⋊C8
C4⋊Q8
C4.6Q16

Character table of C4.6Q16

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1111222248844444444
ρ11111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-11-1-111-1-11    linear of order 2
ρ4111111111-1-1-111-1-111-1    linear of order 2
ρ51111-1-1-1-11-11-i-iii-ii-ii    linear of order 4
ρ61111-1-1-1-11-11ii-i-ii-ii-i    linear of order 4
ρ71111-1-1-1-111-1-ii-ii-i-iii    linear of order 4
ρ81111-1-1-1-111-1i-ii-iii-i-i    linear of order 4
ρ92222-222-2-20000000000    orthogonal lifted from D4
ρ1022222-2-22-20000000000    orthogonal lifted from D4
ρ112-22-2200-20000-√2-√200√2√20    symplectic lifted from Q16, Schur index 2
ρ122-2-220-220000-√200√2√200-√2    symplectic lifted from Q16, Schur index 2
ρ132-2-220-220000√200-√2-√200√2    symplectic lifted from Q16, Schur index 2
ρ142-22-2200-20000√2√200-√2-√20    symplectic lifted from Q16, Schur index 2
ρ152-22-2-20020000-√-2√-200-√-2√-20    complex lifted from SD16
ρ162-22-2-20020000√-2-√-200√-2-√-20    complex lifted from SD16
ρ172-2-2202-20000√-200√-2-√-200-√-2    complex lifted from SD16
ρ182-2-2202-20000-√-200-√-2√-200√-2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C4.D4

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

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

C4.6Q16 is a maximal subgroup of
 C42.4D4  C42.410D4  C42.411D4  C42.415D4  C42.79D4  C42.80D4  C42.418D4  C42.85D4  C42.86D4  C42.87D4  D4⋊SD16  Q8⋊SD16  C42.185C23  D4⋊Q16  Q8⋊Q16  C42.195C23  D4.5SD16  D4⋊3Q16  Q8⋊3Q16  C42.207C23  C8⋊11SD16  C8⋊8Q16  D4.1Q16  D4.2SD16  Q8.2SD16  D4.Q16  Q8.2Q16  C8⋊3SD16  C8⋊Q16  C8.8SD16  C42.248C23  C42.249C23  C42.254C23  C42.255C23  Dic5.Q16
 C4p.Q16: Q8.1Q16  C8.3Q16  C4.Dic12  C12.5Q16  C4.Dic20  C20.5Q16  C4.Dic28  C28.5Q16 ...
C4.6Q16 is a maximal quotient of
 (C2×Q8)⋊C8  C42.8Q8  Dic5.Q16
 C4p.Q16: C8.1Q16  C4.Dic12  C12.5Q16  C4.Dic20  C20.5Q16  C4.Dic28  C28.5Q16 ...

Matrix representation of C4.6Q16 ►in GL4(𝔽17) generated by

0100
16000
00160
00016
,
101600
16700
00512
0055
,
1700
71600
00101
0017
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,5,5,0,0,12,5],[1,7,0,0,7,16,0,0,0,0,10,1,0,0,1,7] >;
 

C4.6Q16 in GAP, Magma, Sage, TeX 

C_4._6Q_{16}
 
% in TeX
 
G:=Group("C4.6Q16");
 
// GroupNames label
 
G:=SmallGroup(64,14);
 
// by ID
 
G=gap.SmallGroup(64,14);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,199,362,332,158,681,165]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=1,c^2=a^2*b^4,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.6Q16 in TeX
Character table of C4.6Q16 in TeX

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