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G = C22.4Q16order 64 = 26

1st central extension by C22 of Q16

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

Aliases: C4.2C42, C22.7D8, C23.52D4, C22.4Q16, C22.9SD16, C4⋊C44C4, (C2×C8)⋊4C4, C4.3(C4⋊C4), (C2×C4).12Q8, (C2×C4).111D4, (C22×C8).2C2, C2.2(C4.Q8), C2.2(C2.D8), C2.2(D4⋊C4), C22.15(C4⋊C4), C4.19(C22⋊C4), C2.2(Q8⋊C4), C22.25(C22⋊C4), C2.5(C2.C42), (C22×C4).102C22, (C2×C4⋊C4).2C2, (C2×C4).39(C2×C4), SmallGroup(64,21)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22.4Q16
C1C2C4C2×C4C22×C4C22×C8 — C22.4Q16
C1C2C4 — C22.4Q16
C1C23C22×C4 — C22.4Q16
C1C2C2C22×C4 — C22.4Q16

Generators and relations for C22.4Q16
 G = < a,b,c,d | a2=b2=c8=1, d2=bc4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Subgroups: 93 in 57 conjugacy classes, 37 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C2×C4⋊C4 [×2], C22×C8, C22.4Q16
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16

Character table of C22.4Q16

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111111122224444444422222222
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ3111111111111-1111-11-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111111-1-1-11-111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1-11-1111-1-1ii-i-iii-i-i-11-1111-1-1    linear of order 4
ρ61-11-1-11-1111-1-1-ii-i-i-iiii1-11-1-1-111    linear of order 4
ρ71-1-1-1111-11-11-11i-ii-1-i-11ii-ii-i-ii-i    linear of order 4
ρ811-11-11-1-11-1-11-i11-1i-1-ii-iiii-i-i-ii    linear of order 4
ρ91-1-1-1111-11-11-1-1-ii-i1i1-1ii-ii-i-ii-i    linear of order 4
ρ1011-11-11-1-11-1-11i11-1-i-1i-ii-i-i-iiii-i    linear of order 4
ρ1111-11-11-1-11-1-11-i-1-11i1-iii-i-i-iiii-i    linear of order 4
ρ121-1-1-1111-11-11-1-1i-ii1-i1-1-i-ii-iii-ii    linear of order 4
ρ1311-11-11-1-11-1-11i-1-11-i1i-i-iiii-i-i-ii    linear of order 4
ρ141-1-1-1111-11-11-11-ii-i-1i-11-i-ii-iii-ii    linear of order 4
ρ151-11-1-11-1111-1-1i-iiii-i-i-i1-11-1-1-111    linear of order 4
ρ161-11-1-11-1111-1-1-i-iii-i-iii-11-1111-1-1    linear of order 4
ρ1722-22-22-2-2-222-20000000000000000    orthogonal lifted from D4
ρ182-2-2-2222-2-22-220000000000000000    orthogonal lifted from D4
ρ1922222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ202-2222-2-2-20000000000002-2-22-22-22    orthogonal lifted from D8
ρ212-2222-2-2-2000000000000-222-22-22-2    orthogonal lifted from D8
ρ22222-2-2-22-200000000000022-2-22-2-22    symplectic lifted from Q16, Schur index 2
ρ232-22-2-22-22-2-2220000000000000000    symplectic lifted from Q8, Schur index 2
ρ24222-2-2-22-2000000000000-2-222-222-2    symplectic lifted from Q16, Schur index 2
ρ2522-2-22-2-22000000000000--2-2--2--2--2-2-2-2    complex lifted from SD16
ρ262-2-22-2-222000000000000-2-2-2--2--2-2--2--2    complex lifted from SD16
ρ2722-2-22-2-22000000000000-2--2-2-2-2--2--2--2    complex lifted from SD16
ρ282-2-22-2-222000000000000--2--2--2-2-2--2-2-2    complex lifted from SD16

Smallest permutation representation of C22.4Q16
Regular action on 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(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 23 31 50)(2 64 32 39)(3 21 25 56)(4 62 26 37)(5 19 27 54)(6 60 28 35)(7 17 29 52)(8 58 30 33)(9 18 41 53)(10 59 42 34)(11 24 43 51)(12 57 44 40)(13 22 45 49)(14 63 46 38)(15 20 47 55)(16 61 48 36)

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

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

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

C22.4Q16 is a maximal subgroup of
C24.132D4  C24.152D4  C4×D4⋊C4  C4×Q8⋊C4  D4⋊C42  Q8⋊C42  C4×C4.Q8  C4×C2.D8  C8⋊C42  C42.98D4  C42.99D4  C42.100D4  C42.101D4  C24.133D4  C24.67D4  C24.157D4  C24.69D4  C42.55Q8  C42.56Q8  C42.24Q8  C24.159D4  C24.71D4  C2.(C4×D8)  Q8⋊(C4⋊C4)  D4⋊(C4⋊C4)  Q8⋊C4⋊C4  C24.160D4  C24.73D4  C24.74D4  (C2×SD16)⋊14C4  (C2×C4)⋊9Q16  (C2×C4)⋊9D8  (C2×SD16)⋊15C4  C24.135D4  C24.75D4  C24.76D4  C2.D84C4  C4.Q89C4  C4.Q810C4  C2.D85C4  D4⋊C4⋊C4  C4.67(C4×D4)  C4.68(C4×D4)  C2.(C4×Q16)  C2.(C88D4)  C2.(C87D4)  C2.(C8⋊D4)  C2.(C82D4)  C42.29Q8  C42.30Q8  C42.31Q8  C42.121D4  C42.122D4  C42.123D4  C42.436D4  C42.125D4  C232D8  C233SD16  C232Q16  (C2×D4)⋊Q8  (C2×Q8)⋊Q8  C4⋊C4.84D4  C4⋊C4.85D4  C24.84D4  C24.85D4  C24.86D4  (C2×C4)⋊3D8  (C2×C4)⋊5SD16  (C2×C4)⋊3Q16  C4⋊C4⋊Q8  (C2×C8)⋊Q8  C4⋊C4.106D4  (C2×Q8).8Q8  (C2×C4).23D8  (C2×C8).52D4  (C2×C4).24D8  (C2×C4).19Q16  C428C4⋊C2  (C2×Q8).109D4  C24.88D4  C24.89D4  (C2×C4).26D8  (C2×C4).21Q16  C4.(C4⋊Q8)  (C2×C4).28D8  (C2×C4).23Q16  C4⋊C4.Q8  D10.18D8  D10.10D8
 C23.D4p: C22.SD32  C23.32D8  C23.22D8  C23.36D8  C23.37D8  C23.38D8  C23.23D8  C24.83D4 ...
 (C2×C4p).Q8: C87(C4⋊C4)  C85(C4⋊C4)  C4.(C4×Q8)  C8⋊(C4⋊C4)  C42.437D4  C42.124D4  C2.(C8⋊Q8)  (C2×C8).1Q8 ...
C22.4Q16 is a maximal quotient of
C42.385D4  C42.46Q8  C42.5Q8  C23.8D8  C42.27D4  C23.30D8  C42.8Q8  C42.389D4  C42.10Q8
 C4p.C42: C8.7C42  C8.8C42  C8.9C42  C8.11C42  C23.9D8  C8.13C42  C8.C42  C8.2C42 ...

Matrix representation of C22.4Q16 in GL4(𝔽17) generated by

16000
01600
00160
00016
,
16000
0100
0010
0001
,
4000
0400
00125
001212
,
13000
0100
00125
0055
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,12,12,0,0,5,12],[13,0,0,0,0,1,0,0,0,0,12,5,0,0,5,5] >;

C22.4Q16 in GAP, Magma, Sage, TeX

C_2^2._4Q_{16}
% in TeX

G:=Group("C2^2.4Q16");
// GroupNames label

G:=SmallGroup(64,21);
// by ID

G=gap.SmallGroup(64,21);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,1444,88]);
// Polycyclic

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

Export

Character table of C22.4Q16 in TeX

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