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G = C22.7C42order 64 = 26

2nd central extension by C22 of C42

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C22.7C42, C22.7M4(2), (C2×C4)⋊2C8, (C2×C8)⋊3C4, C2.2(C4×C8), C2.1(C4⋊C8), C4.18(C4⋊C4), (C2×C4).23Q8, (C2×C4).140D4, (C22×C4).7C4, (C22×C8).1C2, (C2×C42).1C2, C22.6(C2×C8), C2.2(C8⋊C4), C2.1(C22⋊C8), C23.36(C2×C4), C4.26(C22⋊C4), C22.14(C4⋊C4), C22.23(C22⋊C4), C2.1(C2.C42), (C22×C4).134C22, (C2×C4).79(C2×C4), SmallGroup(64,17)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22.7C42
C1C2C22C2×C4C22×C4C2×C42 — C22.7C42
C1C2 — C22.7C42
C1C22×C4 — C22.7C42
C1C2C2C22×C4 — C22.7C42

Generators and relations for C22.7C42
 G = < a,b,c,d | a2=b2=d4=1, c4=b, ab=ba, dcd-1=ac=ca, ad=da, bc=cb, bd=db >

Subgroups: 77 in 59 conjugacy classes, 41 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×C42, C22×C8 [×2], C22.7C42
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42

Smallest permutation representation of C22.7C42
Regular action on 64 points
Generators in S64
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(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 63 55 46)(2 38 56 22)(3 57 49 48)(4 40 50 24)(5 59 51 42)(6 34 52 18)(7 61 53 44)(8 36 54 20)(9 64 32 47)(10 39 25 23)(11 58 26 41)(12 33 27 17)(13 60 28 43)(14 35 29 19)(15 62 30 45)(16 37 31 21)

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

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

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

C22.7C42 is a maximal subgroup of
(C2×C4).98D8  C4⋊C4⋊C8  (C2×Q8)⋊C8  C23.28C42  C23.29C42  C43.C2  (C4×C8)⋊12C4  C4×C22⋊C8  C42.378D4  C42.379D4  C8×C22⋊C4  C23.36C42  C23.17C42  C4×C4⋊C8  C43.7C2  C42.45Q8  C8×C4⋊C4  C4⋊C813C4  C4⋊C814C4  C42.425D4  C42.95D4  C23.32M4(2)  C24.53(C2×C4)  C428C8  C42.23Q8  C425C8  C424C4.C2  C23.21M4(2)  (C2×C8).195D4  C2.(C4×D8)  Q8⋊(C4⋊C4)  D4⋊(C4⋊C4)  Q8⋊C4⋊C4  M4(2).42D4  C23.22M4(2)  C232M4(2)  M4(2).43D4  (C2×SD16)⋊14C4  (C2×C4)⋊9Q16  (C2×C4)⋊9D8  (C2×SD16)⋊15C4  C4⋊C43C8  (C2×C8).Q8  C2.D84C4  C4.Q89C4  C4.Q810C4  C2.D85C4  M4(2).3Q8  C22⋊C44C8  C23.9M4(2)  D4⋊C4⋊C4  C4.67(C4×D4)  C4.68(C4×D4)  C2.(C4×Q16)  M4(2).24D4  C42.61Q8  C42.27Q8  C42.327D4  C42.120D4  (C2×C4)⋊2D8  (C22×D8).C2  (C2×C4)⋊3SD16  (C2×C8)⋊20D4  (C2×C8).41D4  (C2×C4)⋊2Q16  (C2×D4)⋊Q8  (C2×Q8)⋊Q8  C4⋊C4.84D4  C4⋊C4.85D4  C4⋊C47D4  C4⋊C4.94D4  C4⋊C4.95D4  C4⋊C4⋊Q8  (C2×C8)⋊Q8  C2.(C8⋊Q8)  (C2×C4).24D8  (C2×C4).19Q16  C428C4⋊C2  (C2×Q8).109D4  (C2×C8).1Q8  C2.(C83Q8)  (C2×C8).24Q8  (C2×C8).168D4  (C2×C4).27D8  (C2×C8).169D4  (C2×C8).60D4  (C2×C8).170D4  (C2×C8).171D4  (C2×C4).28D8  (C2×C4).23Q16  C4⋊C4.Q8
 C2p.(C4×C8): C424C8  (C2×C12)⋊3C8  (C2×C24)⋊5C4  (C2×C20)⋊8C8  (C2×C40)⋊15C4  D10.3M4(2)  C10.(C4⋊C8)  (C2×C28)⋊3C8 ...
C22.7C42 is a maximal quotient of
C421C8  C426C8  M4(2)⋊C8  C42.46Q8  C23.19C42  C23.21C42  C42.3Q8  C42.7C8  M5(2)⋊C4
 (C2×C4p)⋊C8: C2.C82  C42.20D4  C42.385D4  C42.2Q8  (C2×C12)⋊3C8  (C2×C20)⋊8C8  C10.(C4⋊C8)  (C2×C28)⋊3C8 ...
 (C2×C8p)⋊C4: C22.7M5(2)  C42.2C8  M4(2).C8  M5(2)⋊7C4  (C2×C24)⋊5C4  (C2×C40)⋊15C4  D10.3M4(2)  (C2×C56)⋊5C4 ...

40 conjugacy classes

class 1 2A···2G4A···4H4I···4P8A···8P
order12···24···44···48···8
size11···11···12···22···2

40 irreducible representations

dim111111222
type++++-
imageC1C2C2C4C4C8D4Q8M4(2)
kernelC22.7C42C2×C42C22×C8C2×C8C22×C4C2×C4C2×C4C2×C4C22
# reps1128416314

Matrix representation of C22.7C42 in GL4(𝔽17) generated by

1000
0100
00160
00016
,
16000
0100
00160
00016
,
15000
0100
00915
00138
,
13000
01300
0041
00013
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[15,0,0,0,0,1,0,0,0,0,9,13,0,0,15,8],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,1,13] >;

C22.7C42 in GAP, Magma, Sage, TeX

C_2^2._7C_4^2
% in TeX

G:=Group("C2^2.7C4^2");
// GroupNames label

G:=SmallGroup(64,17);
// by ID

G=gap.SmallGroup(64,17);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,117]);
// Polycyclic

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

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