p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊9C4, C23.60C23, C4⋊1(C4⋊C4), (C2×C4).67D4, C2.1(C4⋊Q8), (C2×C4).14Q8, (C2×C42).9C2, C2.1(C4⋊1D4), C22.33(C2×D4), C22.11(C2×Q8), C22.33(C22×C4), (C22×C4).104C22, C2.6(C2×C4⋊C4), (C2×C4⋊C4).5C2, (C2×C4).70(C2×C4), SmallGroup(64,65)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊9C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 129 in 93 conjugacy classes, 65 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2×C42, C2×C4⋊C4, C42⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C42⋊9C4
Character table of C42⋊9C4
| 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 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
►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 13 51 40)(2 14 52 37)(3 15 49 38)(4 16 50 39)(5 44 55 20)(6 41 56 17)(7 42 53 18)(8 43 54 19)(9 31 36 28)(10 32 33 25)(11 29 34 26)(12 30 35 27)(21 61 45 60)(22 62 46 57)(23 63 47 58)(24 64 48 59)
(1 63 29 56)(2 62 30 55)(3 61 31 54)(4 64 32 53)(5 52 57 27)(6 51 58 26)(7 50 59 25)(8 49 60 28)(9 19 38 45)(10 18 39 48)(11 17 40 47)(12 20 37 46)(13 23 34 41)(14 22 35 44)(15 21 36 43)(16 24 33 42)
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), (1,13,51,40)(2,14,52,37)(3,15,49,38)(4,16,50,39)(5,44,55,20)(6,41,56,17)(7,42,53,18)(8,43,54,19)(9,31,36,28)(10,32,33,25)(11,29,34,26)(12,30,35,27)(21,61,45,60)(22,62,46,57)(23,63,47,58)(24,64,48,59), (1,63,29,56)(2,62,30,55)(3,61,31,54)(4,64,32,53)(5,52,57,27)(6,51,58,26)(7,50,59,25)(8,49,60,28)(9,19,38,45)(10,18,39,48)(11,17,40,47)(12,20,37,46)(13,23,34,41)(14,22,35,44)(15,21,36,43)(16,24,33,42)>;
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), (1,13,51,40)(2,14,52,37)(3,15,49,38)(4,16,50,39)(5,44,55,20)(6,41,56,17)(7,42,53,18)(8,43,54,19)(9,31,36,28)(10,32,33,25)(11,29,34,26)(12,30,35,27)(21,61,45,60)(22,62,46,57)(23,63,47,58)(24,64,48,59), (1,63,29,56)(2,62,30,55)(3,61,31,54)(4,64,32,53)(5,52,57,27)(6,51,58,26)(7,50,59,25)(8,49,60,28)(9,19,38,45)(10,18,39,48)(11,17,40,47)(12,20,37,46)(13,23,34,41)(14,22,35,44)(15,21,36,43)(16,24,33,42) );
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)], [(1,13,51,40),(2,14,52,37),(3,15,49,38),(4,16,50,39),(5,44,55,20),(6,41,56,17),(7,42,53,18),(8,43,54,19),(9,31,36,28),(10,32,33,25),(11,29,34,26),(12,30,35,27),(21,61,45,60),(22,62,46,57),(23,63,47,58),(24,64,48,59)], [(1,63,29,56),(2,62,30,55),(3,61,31,54),(4,64,32,53),(5,52,57,27),(6,51,58,26),(7,50,59,25),(8,49,60,28),(9,19,38,45),(10,18,39,48),(11,17,40,47),(12,20,37,46),(13,23,34,41),(14,22,35,44),(15,21,36,43),(16,24,33,42)]])
C42⋊9C4 is a maximal subgroup of
C42.8Q8 C42⋊3C8 C42.98D4 C42.99D4 C42.55Q8 C42.24Q8 C4≀C2⋊C4 C42.29Q8 C42.30Q8 C42.431D4 C42.432D4 C42.110D4 C42.436D4 C42.124D4 C42⋊11D4 M4(2)⋊Q8 (C2×C4).24D8 (C2×C4).19Q16 (C2×C8).1Q8 C2.(C8⋊3Q8) (C2×C4).27D8 (C2×C8).169D4 (C2×C8).60D4 (C2×C8).170D4 (C2×C4).28D8 (C2×C4).23Q16 C23.167C24 C4×C4⋊1D4 C4×C4⋊Q8 C24.192C23 C23.199C24 C42.160D4 C42.33Q8 D4×C4⋊C4 Q8×C4⋊C4 C23.236C24 C23.237C24 C24.230C23 C23.322C24 C23.323C24 C24.568C23 C24.268C23 C23.396C24 C23.397C24 C24.308C23 C23.400C24 C23.401C24 C23.402C24 C23.406C24 C23.407C24 C23.411C24 C23.412C24 C42⋊18D4 C42.166D4 C42.167D4 C42⋊7Q8 C42.35Q8 C42.174D4 C42.175D4 C42.176D4 C42.36Q8 C42.37Q8 C42.180D4 C42⋊28D4 C42.188D4 C42.39Q8 C42⋊10Q8 C23.580C24 C23.618C24 C23.620C24 C23.621C24 C24.454C23 C23.691C24 C23.692C24 C23.693C24 C23.694C24 C23.695C24 C42⋊35D4 C42⋊12Q8 C42⋊47D4 C42.440D4 C43.15C2 C42⋊19Q8 C42⋊2C12
C4p⋊(C4⋊C4): C42.58Q8 C42.59Q8 C42.26Q8 C42⋊10Dic3 (C4×Dic3)⋊8C4 C42⋊8Dic5 C20⋊5(C4⋊C4) C42⋊8F5 ...
C42⋊9C4 is a maximal quotient of
C24.625C23 C24.634C23 C42⋊9C8 C42.25Q8 C42.60Q8 C42.324D4 C42.106D4
C4p⋊(C4⋊C4): C42.58Q8 C42.59Q8 C42.26Q8 C42⋊10Dic3 (C4×Dic3)⋊8C4 C42⋊8Dic5 C20⋊5(C4⋊C4) C42⋊8F5 ...
Matrix representation of C42⋊9C4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 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))| [4,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3],[4,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4],[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⋊9C4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_9C_4
% in TeX
G:=Group("C4^2:9C4"); // GroupNames label
G:=SmallGroup(64,65);
// by ID
G=gap.SmallGroup(64,65);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,55,362,86]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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