p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊5C4, C23.59C23, (C2×C42).3C2, (C22×C4).3C22, C2.8(C42⋊C2), C2.1(C42⋊2C2), C22.18(C4○D4), C2.C42.2C2, C22.32(C22×C4), (C2×C4).54(C2×C4), SmallGroup(64,64)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊5C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, cbc-1=a2b-1 >
Subgroups: 105 in 69 conjugacy classes, 41 normal (5 characteristic)
C1, C2, C4, C22, C22, C2×C4, C2×C4, C23, C42, C22×C4, C2.C42, C2×C42, C42⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C42⋊2C2, C42⋊5C4
Character table of C42⋊5C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►Regular action on 64 points
(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 28 31 40)(2 25 32 37)(3 26 29 38)(4 27 30 39)(5 55 22 48)(6 56 23 45)(7 53 24 46)(8 54 21 47)(9 51 16 36)(10 52 13 33)(11 49 14 34)(12 50 15 35)(17 58 63 43)(18 59 64 44)(19 60 61 41)(20 57 62 42)
(1 47 11 63)(2 55 12 18)(3 45 9 61)(4 53 10 20)(5 52 44 27)(6 34 41 40)(7 50 42 25)(8 36 43 38)(13 62 30 46)(14 17 31 54)(15 64 32 48)(16 19 29 56)(21 51 58 26)(22 33 59 39)(23 49 60 28)(24 35 57 37)
G:=sub<Sym(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), (1,28,31,40)(2,25,32,37)(3,26,29,38)(4,27,30,39)(5,55,22,48)(6,56,23,45)(7,53,24,46)(8,54,21,47)(9,51,16,36)(10,52,13,33)(11,49,14,34)(12,50,15,35)(17,58,63,43)(18,59,64,44)(19,60,61,41)(20,57,62,42), (1,47,11,63)(2,55,12,18)(3,45,9,61)(4,53,10,20)(5,52,44,27)(6,34,41,40)(7,50,42,25)(8,36,43,38)(13,62,30,46)(14,17,31,54)(15,64,32,48)(16,19,29,56)(21,51,58,26)(22,33,59,39)(23,49,60,28)(24,35,57,37)>;
G:=Group( (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), (1,28,31,40)(2,25,32,37)(3,26,29,38)(4,27,30,39)(5,55,22,48)(6,56,23,45)(7,53,24,46)(8,54,21,47)(9,51,16,36)(10,52,13,33)(11,49,14,34)(12,50,15,35)(17,58,63,43)(18,59,64,44)(19,60,61,41)(20,57,62,42), (1,47,11,63)(2,55,12,18)(3,45,9,61)(4,53,10,20)(5,52,44,27)(6,34,41,40)(7,50,42,25)(8,36,43,38)(13,62,30,46)(14,17,31,54)(15,64,32,48)(16,19,29,56)(21,51,58,26)(22,33,59,39)(23,49,60,28)(24,35,57,37) );
G=PermutationGroup([[(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)], [(1,28,31,40),(2,25,32,37),(3,26,29,38),(4,27,30,39),(5,55,22,48),(6,56,23,45),(7,53,24,46),(8,54,21,47),(9,51,16,36),(10,52,13,33),(11,49,14,34),(12,50,15,35),(17,58,63,43),(18,59,64,44),(19,60,61,41),(20,57,62,42)], [(1,47,11,63),(2,55,12,18),(3,45,9,61),(4,53,10,20),(5,52,44,27),(6,34,41,40),(7,50,42,25),(8,36,43,38),(13,62,30,46),(14,17,31,54),(15,64,32,48),(16,19,29,56),(21,51,58,26),(22,33,59,39),(23,49,60,28),(24,35,57,37)]])
C42⋊5C4 is a maximal subgroup of
C23.165C24 C4×C42⋊2C2 C24.547C23 C23.202C24 C23.218C24 C24.205C23 C23.225C24 C23.229C24 C23.235C24 C23.238C24 C23.262C24 C23.263C24 C24.230C23 C24.577C23 C23.395C24 C23.410C24 C23.414C24 C24.309C23 C23.418C24 C24.313C23 C23.424C24 C23.425C24 C23.432C24 C23.433C24 C23.472C24 C23.473C24 C24.341C23 C23.478C24 C42⋊22D4 C42.183D4 C42⋊8Q8 C42.38Q8 C42⋊29D4 C42.189D4 C42.190D4 C42.191D4 C42⋊11Q8 C23.637C24 C24.426C23 C23.649C24 C23.656C24 C24.438C23 C23.658C24 C23.659C24 C23.662C24 C23.664C24 C23.666C24 C23.669C24 C24.445C23 C23.675C24 C23.676C24 C23.677C24 C42⋊33D4 C42.200D4 C42⋊12Q8 C42⋊43D4 C42⋊15Q8 C42⋊C12
(C4×C4p)⋊C4: C42.6Q8 C42⋊7Dic3 C42⋊5Dic5 C42⋊5F5 C42⋊5Dic7 ...
(C22×C4).D2p: C42.59D4 C42.60D4 C42.63D4 C3⋊(C42⋊5C4) C5⋊2(C42⋊5C4) C7⋊(C42⋊5C4) ...
C42⋊5C4 is a maximal quotient of
C42⋊5C8 C42⋊4C4.C2
(C4×C4p)⋊C4: C8⋊C4⋊17C4 C42⋊7Dic3 C42⋊5Dic5 C42⋊5F5 C42⋊5Dic7 ...
(C22×C4).D2p: C24.624C23 C24.633C23 C3⋊(C42⋊5C4) C5⋊2(C42⋊5C4) C7⋊(C42⋊5C4) ...
Matrix representation of C42⋊5C4 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,3],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,0,3],[2,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0] >;
C42⋊5C4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5C_4
% in TeX
G:=Group("C4^2:5C4"); // GroupNames label
G:=SmallGroup(64,64);
// by ID
G=gap.SmallGroup(64,64);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,151,362,50]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a^2*b^-1>;
// generators/relations
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