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G = C42Q16order 64 = 26

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

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

Aliases: C42Q16, 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

C1C2×C4 — C42Q16
C1C2C4C2×C4C2×Q8C4×Q8 — C42Q16
C1C2C2×C4 — C42Q16
C1C22C42 — C42Q16
C1C2C2C2×C4 — C42Q16

Generators and relations for C42Q16
 G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

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

Character table of C42Q16

 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
ρ132-2-22200-20000000-2-222    symplectic lifted from Q16, Schur index 2
ρ142-2-22200-2000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ152-2-22-200200000002-22-2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-20020000000-22-22    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 C42Q16
Regular action on 64 points
Generators in S64
(1 42 19 30)(2 31 20 43)(3 44 21 32)(4 25 22 45)(5 46 23 26)(6 27 24 47)(7 48 17 28)(8 29 18 41)(9 33 63 51)(10 52 64 34)(11 35 57 53)(12 54 58 36)(13 37 59 55)(14 56 60 38)(15 39 61 49)(16 50 62 40)
(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 37 5 33)(2 36 6 40)(3 35 7 39)(4 34 8 38)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 60 45 64)(42 59 46 63)(43 58 47 62)(44 57 48 61)

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

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

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

C42Q16 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: C88Q16  C87Q16  C8⋊Q16  C82Q16  C4⋊Dic12  C127Q16  C12⋊Q16  C4⋊Dic20 ...
 (Cp×Q8).D4: C42.201C23  Q8.D8  D43Q16  Q83Q16  Q84Q16  D44Q16  C42.213C23  Q8.SD16 ...
 C4⋊C4.D2p: C42.19C23  C42.354C23  C42.361C23  C42.409C23  C42.411C23  C42.25C23  C42.28C23  SD168D4 ...
C42Q16 is a maximal quotient of
C2.(C4×Q16)  C42.29Q8  C42.117D4  (C2×C8).1Q8
 C4p⋊Q16: C88Q16  C87Q16  C8⋊Q16  C82Q16  C4⋊Dic12  C127Q16  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 C42Q16 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] >;

C42Q16 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 C42Q16 in TeX
Character table of C42Q16 in TeX

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