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
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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C2×C42, C22×C8, C22.7C42
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42
►Regular action on 64 points
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(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 11)(2 38 56 47)(3 57 49 13)(4 40 50 41)(5 59 51 15)(6 34 52 43)(7 61 53 9)(8 36 54 45)(10 22 62 30)(12 24 64 32)(14 18 58 26)(16 20 60 28)(17 39 25 48)(19 33 27 42)(21 35 29 44)(23 37 31 46)
G:=sub<Sym(64)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(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), (1,63,55,11)(2,38,56,47)(3,57,49,13)(4,40,50,41)(5,59,51,15)(6,34,52,43)(7,61,53,9)(8,36,54,45)(10,22,62,30)(12,24,64,32)(14,18,58,26)(16,20,60,28)(17,39,25,48)(19,33,27,42)(21,35,29,44)(23,37,31,46)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(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), (1,63,55,11)(2,38,56,47)(3,57,49,13)(4,40,50,41)(5,59,51,15)(6,34,52,43)(7,61,53,9)(8,36,54,45)(10,22,62,30)(12,24,64,32)(14,18,58,26)(16,20,60,28)(17,39,25,48)(19,33,27,42)(21,35,29,44)(23,37,31,46) );
G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(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)], [(1,63,55,11),(2,38,56,47),(3,57,49,13),(4,40,50,41),(5,59,51,15),(6,34,52,43),(7,61,53,9),(8,36,54,45),(10,22,62,30),(12,24,64,32),(14,18,58,26),(16,20,60,28),(17,39,25,48),(19,33,27,42),(21,35,29,44),(23,37,31,46)]])
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⋊C8⋊13C4 C4⋊C8⋊14C4 C42.425D4 C42.95D4 C23.32M4(2) C24.53(C2×C4) C42⋊8C8 C42.23Q8 C42⋊5C8 C42⋊4C4.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) C23⋊2M4(2) M4(2).43D4 (C2×SD16)⋊14C4 (C2×C4)⋊9Q16 (C2×C4)⋊9D8 (C2×SD16)⋊15C4 C4⋊C4⋊3C8 (C2×C8).Q8 C2.D8⋊4C4 C4.Q8⋊9C4 C4.Q8⋊10C4 C2.D8⋊5C4 M4(2).3Q8 C22⋊C4⋊4C8 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⋊C4⋊7D4 C4⋊C4.94D4 C4⋊C4.95D4 C4⋊C4⋊Q8 (C2×C8)⋊Q8 C2.(C8⋊Q8) (C2×C4).24D8 (C2×C4).19Q16 C42⋊8C4⋊C2 (C2×Q8).109D4 (C2×C8).1Q8 C2.(C8⋊3Q8) (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): C42⋊4C8 (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
C42⋊1C8 C42⋊6C8 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 | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C22.7C42 | C2×C42 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 | 3 | 1 | 4 |
Matrix representation of C22.7C42 ►in GL4(𝔽17) generated by
| 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 | 15 |
| 0 | 0 | 13 | 8 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 13 |
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