p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊4C4, C4.7C42, C23.52C23, C2.2(C2×C42), (C2×C42).2C2, C4○(C2.C42), C2.1(C42⋊C2), C22.12(C4○D4), C22.14(C22×C4), (C22×C4).84C22, C2.C42.11C2, (C2×C4).51(C2×C4), SmallGroup(64,57)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊4C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, bc=cb >
Subgroups: 113 in 89 conjugacy classes, 65 normal (5 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C23, C42, C22×C4, C22×C4, C2.C42, C2×C42, C42⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C2×C42, C42⋊C2, C42⋊4C4
►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 7 20 54)(2 8 17 55)(3 5 18 56)(4 6 19 53)(9 31 47 38)(10 32 48 39)(11 29 45 40)(12 30 46 37)(13 43 59 27)(14 44 60 28)(15 41 57 25)(16 42 58 26)(21 52 34 62)(22 49 35 63)(23 50 36 64)(24 51 33 61)
(1 44 49 9)(2 25 50 48)(3 42 51 11)(4 27 52 46)(5 58 33 29)(6 13 34 37)(7 60 35 31)(8 15 36 39)(10 17 41 64)(12 19 43 62)(14 22 38 54)(16 24 40 56)(18 26 61 45)(20 28 63 47)(21 30 53 59)(23 32 55 57)
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), (1,7,20,54)(2,8,17,55)(3,5,18,56)(4,6,19,53)(9,31,47,38)(10,32,48,39)(11,29,45,40)(12,30,46,37)(13,43,59,27)(14,44,60,28)(15,41,57,25)(16,42,58,26)(21,52,34,62)(22,49,35,63)(23,50,36,64)(24,51,33,61), (1,44,49,9)(2,25,50,48)(3,42,51,11)(4,27,52,46)(5,58,33,29)(6,13,34,37)(7,60,35,31)(8,15,36,39)(10,17,41,64)(12,19,43,62)(14,22,38,54)(16,24,40,56)(18,26,61,45)(20,28,63,47)(21,30,53,59)(23,32,55,57)>;
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), (1,7,20,54)(2,8,17,55)(3,5,18,56)(4,6,19,53)(9,31,47,38)(10,32,48,39)(11,29,45,40)(12,30,46,37)(13,43,59,27)(14,44,60,28)(15,41,57,25)(16,42,58,26)(21,52,34,62)(22,49,35,63)(23,50,36,64)(24,51,33,61), (1,44,49,9)(2,25,50,48)(3,42,51,11)(4,27,52,46)(5,58,33,29)(6,13,34,37)(7,60,35,31)(8,15,36,39)(10,17,41,64)(12,19,43,62)(14,22,38,54)(16,24,40,56)(18,26,61,45)(20,28,63,47)(21,30,53,59)(23,32,55,57) );
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)], [(1,7,20,54),(2,8,17,55),(3,5,18,56),(4,6,19,53),(9,31,47,38),(10,32,48,39),(11,29,45,40),(12,30,46,37),(13,43,59,27),(14,44,60,28),(15,41,57,25),(16,42,58,26),(21,52,34,62),(22,49,35,63),(23,50,36,64),(24,51,33,61)], [(1,44,49,9),(2,25,50,48),(3,42,51,11),(4,27,52,46),(5,58,33,29),(6,13,34,37),(7,60,35,31),(8,15,36,39),(10,17,41,64),(12,19,43,62),(14,22,38,54),(16,24,40,56),(18,26,61,45),(20,28,63,47),(21,30,53,59),(23,32,55,57)]])
C42⋊4C4 is a maximal subgroup of
C42⋊1C8 C42.9Q8 C42.375D4 C42.55D4 C42.56D4 (C4×C8)⋊12C4 C42.45Q8 C42.23Q8 C42⋊4C4.C2 C42.25Q8 C42⋊8D4 C42⋊Q8 C23.165C24 C42⋊42D4 C42⋊14Q8 C43⋊2C2 C23.201C24 C23.202C24 C42.160D4 C42.161D4 C42⋊14D4 C42.33Q8 C42⋊4Q8 C23.225C24 C24.208C23 C23.235C24 C23.238C24 C23.253C24 C24.221C23 C23.255C24 C42⋊16D4 C42.163D4 C42⋊5Q8 C23.301C24 C42.34Q8 C23.426C24 C24.315C23 C23.428C24 C23.429C24 C23.430C24 C23.431C24 C23.432C24 C23.433C24 C42⋊24D4 C42.184D4 C42⋊8Q8 C42.38Q8 C42⋊26D4 C42.185D4 C42⋊9Q8 C42.193D4 C42.194D4 C42.195D4 C23.544C24 C23.545C24 C42.39Q8 C42⋊31D4 C42.196D4 C42⋊10Q8 C42⋊4C4⋊C3
C4p.C42: C2.C43 C42⋊6Dic3 C42⋊4Dic5 C42⋊4F5 C42⋊4Dic7 ...
C2p.(C2×C42): D4.C42 C4×C42⋊C2 D4⋊4C42 Q8⋊4C42 Dic3.5C42 Dic5.15C42 Dic7.5C42 ...
C42⋊4C4 is a maximal quotient of
C42⋊4C8 C43.C2 (C4×C8)⋊12C4
C4p.C42: C8.16C42 C42⋊6Dic3 C42⋊4Dic5 C42⋊4F5 C42⋊4Dic7 ...
(C22×C4).D2p: C4×C2.C42 C24.624C23 Dic3.5C42 Dic5.15C42 Dic7.5C42 ...
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4○D4 |
| kernel | C42⋊4C4 | C2.C42 | C2×C42 | C42 | C22 |
| # reps | 1 | 4 | 3 | 24 | 8 |
Matrix representation of C42⋊4C4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0] >;
C42⋊4C4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4C_4
% in TeX
G:=Group("C4^2:4C4"); // GroupNames label
G:=SmallGroup(64,57);
// by ID
G=gap.SmallGroup(64,57);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,50]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,c*a*c^-1=a*b^2,b*c=c*b>;
// generators/relations