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G = C4⋊Q16order 64 = 26

The semidirect product of C4 and Q16 acting via Q16/C8=C2

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

Aliases: C41Q16, C8.8D4, C42.81C22, (C4×C8).7C2, C4.3(C2×D4), C4⋊Q8.9C2, (C2×C4).78D4, (C2×Q16).3C2, C2.10(C2×Q16), C2.7(C41D4), (C2×C8).79C22, (C2×C4).119C23, C22.115(C2×D4), (C2×Q8).25C22, SmallGroup(64,175)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4⋊Q16
C1C2C22C2×C4C42C4×C8 — C4⋊Q16
C1C2C2×C4 — C4⋊Q16
C1C22C42 — C4⋊Q16
C1C2C2C2×C4 — C4⋊Q16

Generators and relations for C4⋊Q16
 G = < a,b,c | a4=b8=1, c2=b4, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 97 in 61 conjugacy classes, 33 normal (7 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×8], C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×8], C2×Q8 [×4], C4×C8, C4⋊Q8 [×2], C2×Q16 [×4], C4⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, Q16 [×4], C2×D4 [×3], C41D4, C2×Q16 [×2], C4⋊Q16

Character table of C4⋊Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111222222888822222222
ρ11111111111111111111111    trivial
ρ21111-1-11-11-11-11-1-1-11-11-111    linear of order 2
ρ31111-1-11-11-11-1-1111-11-11-1-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-11-11-1-111-111-11-11-1-1    linear of order 2
ρ61111111111-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111111111-1-1-1-111111111    linear of order 2
ρ81111-1-11-11-1-11-11-1-11-11-111    linear of order 2
ρ92-2-2200-20200000220-20-200    orthogonal lifted from D4
ρ102-2-220020-200000002020-2-2    orthogonal lifted from D4
ρ112-2-2200-20200000-2-2020200    orthogonal lifted from D4
ρ122-2-220020-20000000-20-2022    orthogonal lifted from D4
ρ1322222-2-2-2-22000000000000    orthogonal lifted from D4
ρ142222-22-22-2-2000000000000    orthogonal lifted from D4
ρ1522-2-2020-2000000-22-2-222-22    symplectic lifted from Q16, Schur index 2
ρ162-22-220000-200002-22-2-22-22    symplectic lifted from Q16, Schur index 2
ρ1722-2-2020-20000002-222-2-22-2    symplectic lifted from Q16, Schur index 2
ρ182-22-220000-20000-22-222-22-2    symplectic lifted from Q16, Schur index 2
ρ192-22-2-20000200002-2-2-2222-2    symplectic lifted from Q16, Schur index 2
ρ2022-2-20-202000000-222-2-222-2    symplectic lifted from Q16, Schur index 2
ρ2122-2-20-2020000002-2-222-2-22    symplectic lifted from Q16, Schur index 2
ρ222-22-2-2000020000-2222-2-2-22    symplectic lifted from Q16, Schur index 2

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

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

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

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

C4⋊Q16 is a maximal subgroup of
C8.25D8  C8.27D8  (C2×Q8).D4  D4⋊Q16  Q8⋊Q16  C42.195C23  C8.28D8  Q81Q16  C8.D8  C8.SD16  C825C2  C8.9SD16  C42.667C23  C8.2D8  D8.4D4  Q16.4D4  C8.5D8  C4.SD32  C8.14SD16  C165D4  C42.360D4  M4(2)⋊8D4  M4(2).20D4  C42.367D4  C42.389C23  C42.262D4  C42.267D4  C42.268D4  C42.409C23  SD163D4  D813D4  D4×Q16  D8○Q16  C42.527C23  Q86Q16  C42.73C23
 C4p⋊Q16: C85Q16  C84Q16  C83Q16  C124Q16  C123Q16  C204Q16  C203Q16  C284Q16 ...
 C8p.D4: C4⋊Q32  C8.7D8  C24.26D4  C40.26D4  C56.26D4 ...
C4⋊Q16 is a maximal quotient of
 C4p⋊Q16: C85Q16  C84Q16  C83Q16  C124Q16  C123Q16  C204Q16  C203Q16  C284Q16 ...
 (C2×C8).D2p: C8.7Q16  C42.59Q8  C42.431D4  (C2×C4)⋊6Q16  (C2×C4)⋊2Q16  (C2×C8).60D4  C24.26D4  C40.26D4 ...

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

01600
1000
00160
00016
,
0100
16000
0080
00915
,
111300
13600
001215
00135
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,8,9,0,0,0,15],[11,13,0,0,13,6,0,0,0,0,12,13,0,0,15,5] >;

C4⋊Q16 in GAP, Magma, Sage, TeX

C_4\rtimes Q_{16}
% in TeX

G:=Group("C4:Q16");
// GroupNames label

G:=SmallGroup(64,175);
// by ID

G=gap.SmallGroup(64,175);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,86,963,117]);
// Polycyclic

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

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Character table of C4⋊Q16 in TeX

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