p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊8C4, C23.58C23, (C2×C4).66D4, C4.10(C4⋊C4), (C2×C4).13Q8, (C2×C42).8C2, C22.32(C2×D4), C22.10(C2×Q8), C2.1(C4.4D4), C2.1(C42.C2), C2.7(C42⋊C2), C22.17(C4○D4), C2.C42.1C2, (C22×C4).89C22, C22.31(C22×C4), C2.5(C2×C4⋊C4), (C2×C4⋊C4).4C2, (C2×C4).53(C2×C4), SmallGroup(64,63)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊8C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, cbc-1=a2b >
Subgroups: 113 in 77 conjugacy classes, 49 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C42⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C42⋊8C4
Character table of C42⋊8C4
| 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 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 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 |
| ρ20 | 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 |
| ρ21 | 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 |
| 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 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 |
| 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 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 37 22 30)(6 38 23 31)(7 39 24 32)(8 40 21 29)(17 50 61 33)(18 51 62 34)(19 52 63 35)(20 49 64 36)(41 46 58 55)(42 47 59 56)(43 48 60 53)(44 45 57 54)
(1 50 6 42)(2 34 7 60)(3 52 8 44)(4 36 5 58)(9 62 30 48)(10 19 31 54)(11 64 32 46)(12 17 29 56)(13 49 22 41)(14 33 23 59)(15 51 24 43)(16 35 21 57)(18 37 53 26)(20 39 55 28)(25 61 40 47)(27 63 38 45)
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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,22,30)(6,38,23,31)(7,39,24,32)(8,40,21,29)(17,50,61,33)(18,51,62,34)(19,52,63,35)(20,49,64,36)(41,46,58,55)(42,47,59,56)(43,48,60,53)(44,45,57,54), (1,50,6,42)(2,34,7,60)(3,52,8,44)(4,36,5,58)(9,62,30,48)(10,19,31,54)(11,64,32,46)(12,17,29,56)(13,49,22,41)(14,33,23,59)(15,51,24,43)(16,35,21,57)(18,37,53,26)(20,39,55,28)(25,61,40,47)(27,63,38,45)>;
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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,22,30)(6,38,23,31)(7,39,24,32)(8,40,21,29)(17,50,61,33)(18,51,62,34)(19,52,63,35)(20,49,64,36)(41,46,58,55)(42,47,59,56)(43,48,60,53)(44,45,57,54), (1,50,6,42)(2,34,7,60)(3,52,8,44)(4,36,5,58)(9,62,30,48)(10,19,31,54)(11,64,32,46)(12,17,29,56)(13,49,22,41)(14,33,23,59)(15,51,24,43)(16,35,21,57)(18,37,53,26)(20,39,55,28)(25,61,40,47)(27,63,38,45) );
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)], [(1,27,14,10),(2,28,15,11),(3,25,16,12),(4,26,13,9),(5,37,22,30),(6,38,23,31),(7,39,24,32),(8,40,21,29),(17,50,61,33),(18,51,62,34),(19,52,63,35),(20,49,64,36),(41,46,58,55),(42,47,59,56),(43,48,60,53),(44,45,57,54)], [(1,50,6,42),(2,34,7,60),(3,52,8,44),(4,36,5,58),(9,62,30,48),(10,19,31,54),(11,64,32,46),(12,17,29,56),(13,49,22,41),(14,33,23,59),(15,51,24,43),(16,35,21,57),(18,37,53,26),(20,39,55,28),(25,61,40,47),(27,63,38,45)]])
C42⋊8C4 is a maximal subgroup of
C23.165C24 C4×C4.4D4 C4×C42.C2 C24.192C23 C24.547C23 C23.202C24 C42.161D4 C42⋊14D4 C42⋊4Q8 C24.203C23 C23.225C24 C24.208C23 C23.231C24 C23.233C24 C23.234C24 C23.237C24 C23.238C24 C24.212C23 C23.261C24 C23.264C24 C24.254C23 C23.321C24 C24.267C23 C24.569C23 C23.388C24 C24.301C23 C24.304C23 C23.395C24 C23.405C24 C23.408C24 C23.409C24 C23.413C24 C23.414C24 C24.309C23 C23.417C24 C23.418C24 C23.419C24 C23.420C24 C23.424C24 C23.428C24 C23.429C24 C42⋊17D4 C42.165D4 C42⋊6Q8 C24.326C23 C42.173D4 C42.174D4 C42.175D4 C42.177D4 C42.36Q8 C42.37Q8 C24.338C23 C24.339C23 C24.340C23 C42.178D4 C42.179D4 C42⋊25D4 C42⋊9Q8 C42⋊27D4 C42.187D4 C42⋊30D4 C42.192D4 C42.39Q8 C42⋊10Q8 C42⋊11Q8 C24.405C23 C23.606C24 C23.619C24 C24.418C23 C24.430C23 C24.437C23 C23.654C24 C23.655C24 C24.440C23 C23.662C24 C23.663C24 C24.443C23 C23.666C24 C23.667C24 C23.668C24 C23.669C24 C23.672C24 C23.673C24 C23.674C24 C23.675C24 C23.681C24 C23.682C24 C23.683C24 C23.685C24 C23.689C24 C42⋊34D4 C42.199D4 C42.201D4 C42⋊13Q8 C42.40Q8 C42.439D4 C43.15C2 C42⋊18Q8 C43.18C2
(C4×C4p)⋊C4: C42.5Q8 C42.56Q8 C42.60Q8 C42⋊11Dic3 C42⋊9Dic5 C42⋊9F5 C42⋊9Dic7 ...
(C2×C4p).Q8: C42.26Q8 C42.437D4 (C2×C8).24Q8 (C4×Dic3)⋊9C4 C20.48(C4⋊C4) (C4×Dic7)⋊9C4 ...
(C22×C4).D2p: C42.7Q8 C42⋊C8 C42.58D4 C42.61D4 C42.62D4 C42.100D4 C42.101D4 C42.24Q8 ...
C42⋊8C4 is a maximal quotient of
C42⋊8C8 C42.23Q8 C42.322D4 C42.104D4
C4p.(C4⋊C4): C8.(C4⋊C4) C42⋊11Dic3 (C4×Dic3)⋊9C4 C42⋊9Dic5 C20.48(C4⋊C4) C42⋊9F5 C42⋊9Dic7 (C4×Dic7)⋊9C4 ...
(C22×C4).D2p: C24.624C23 C24.625C23 C24.632C23 C42.55Q8 C42.56Q8 C42.24Q8 C3⋊(C42⋊8C4) C5⋊2(C42⋊8C4) ...
Matrix representation of C42⋊8C4 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 1 | 0 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,4,0],[2,0,0,0,0,0,0,2,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,3] >;
C42⋊8C4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8C_4
% in TeX
G:=Group("C4^2:8C4"); // GroupNames label
G:=SmallGroup(64,63);
// by ID
G=gap.SmallGroup(64,63);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,103,362,50]);
// 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>;
// generators/relations
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