direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C4⋊C4, C4⋊C42, C42⋊7C4, C23.54C23, C2.2(C4×D4), C2.1(C4×Q8), (C2×C4).24Q8, (C2×C4).143D4, C2.4(C2×C42), (C2×C42).7C2, C22.8(C2×Q8), C22.28(C2×D4), C4○3(C2.C42), C2.3(C42⋊C2), C22.14(C4○D4), C22.16(C22×C4), (C22×C4).86C22, C2.C42.12C2, C2.2(C2×C4⋊C4), (C2×C4⋊C4).21C2, (C2×C4).33(C2×C4), (C2×C4)○2(C2.C42), SmallGroup(64,59)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4⋊C4
G = < a,b,c | a4=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 121 in 97 conjugacy classes, 73 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×C4⋊C4
►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 50 17 47)(2 51 18 48)(3 52 19 45)(4 49 20 46)(5 43 13 37)(6 44 14 38)(7 41 15 39)(8 42 16 40)(9 33 63 30)(10 34 64 31)(11 35 61 32)(12 36 62 29)(21 60 27 54)(22 57 28 55)(23 58 25 56)(24 59 26 53)
(1 31 21 38)(2 32 22 39)(3 29 23 40)(4 30 24 37)(5 46 9 53)(6 47 10 54)(7 48 11 55)(8 45 12 56)(13 49 63 59)(14 50 64 60)(15 51 61 57)(16 52 62 58)(17 34 27 44)(18 35 28 41)(19 36 25 42)(20 33 26 43)
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), (1,50,17,47)(2,51,18,48)(3,52,19,45)(4,49,20,46)(5,43,13,37)(6,44,14,38)(7,41,15,39)(8,42,16,40)(9,33,63,30)(10,34,64,31)(11,35,61,32)(12,36,62,29)(21,60,27,54)(22,57,28,55)(23,58,25,56)(24,59,26,53), (1,31,21,38)(2,32,22,39)(3,29,23,40)(4,30,24,37)(5,46,9,53)(6,47,10,54)(7,48,11,55)(8,45,12,56)(13,49,63,59)(14,50,64,60)(15,51,61,57)(16,52,62,58)(17,34,27,44)(18,35,28,41)(19,36,25,42)(20,33,26,43)>;
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), (1,50,17,47)(2,51,18,48)(3,52,19,45)(4,49,20,46)(5,43,13,37)(6,44,14,38)(7,41,15,39)(8,42,16,40)(9,33,63,30)(10,34,64,31)(11,35,61,32)(12,36,62,29)(21,60,27,54)(22,57,28,55)(23,58,25,56)(24,59,26,53), (1,31,21,38)(2,32,22,39)(3,29,23,40)(4,30,24,37)(5,46,9,53)(6,47,10,54)(7,48,11,55)(8,45,12,56)(13,49,63,59)(14,50,64,60)(15,51,61,57)(16,52,62,58)(17,34,27,44)(18,35,28,41)(19,36,25,42)(20,33,26,43) );
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)], [(1,50,17,47),(2,51,18,48),(3,52,19,45),(4,49,20,46),(5,43,13,37),(6,44,14,38),(7,41,15,39),(8,42,16,40),(9,33,63,30),(10,34,64,31),(11,35,61,32),(12,36,62,29),(21,60,27,54),(22,57,28,55),(23,58,25,56),(24,59,26,53)], [(1,31,21,38),(2,32,22,39),(3,29,23,40),(4,30,24,37),(5,46,9,53),(6,47,10,54),(7,48,11,55),(8,45,12,56),(13,49,63,59),(14,50,64,60),(15,51,61,57),(16,52,62,58),(17,34,27,44),(18,35,28,41),(19,36,25,42),(20,33,26,43)]])
C4×C4⋊C4 is a maximal subgroup of
C42.46Q8 C42.4Q8 C42.10Q8 C42.403D4 C42.404D4 C42.57D4 C8⋊C42 C42.102D4 C4⋊C4⋊3C8 (C2×C8).Q8 C2.D8⋊4C4 C4.Q8⋊9C4 C4.Q8⋊10C4 C2.D8⋊5C4 C42.61Q8 C42.27Q8 M4(2)⋊7Q8 C42.121D4 C42.122D4 C42.123D4 D4×C42 Q8×C42 C24.524C23 D4⋊4C42 Q8⋊4C42 C23.165C24 C23.167C24 C42⋊14Q8 C23.178C24 C43⋊2C2 C24.192C23 C23.201C24 C23.202C24 C42.33Q8 C42⋊4Q8 C23.214C24 C24.203C23 C24.204C23 C23.218C24 C24.205C23 C23.225C24 C23.226C24 C23.227C24 C24.208C23 C23.229C24 C23.231C24 C23.233C24 C23.237C24 C23.238C24 C24.212C23 C23.241C24 C24.215C23 C24.219C23 C23.250C24 C23.251C24 C23.252C24 C23.253C24 C23.255C24 C24.223C23 C42⋊15D4 C23.295C24 C42.162D4 C42⋊5Q8 C23.301C24 C42.34Q8 C23.345C24 C23.346C24 C24.271C23 C23.348C24 C23.351C24 C23.352C24 C23.353C24 C23.354C24 C24.282C23 C23.362C24 C23.368C24 C23.369C24 C23.374C24 C23.375C24 C24.295C23 C23.379C24 C24.301C23 C23.390C24 C23.391C24 C23.392C24 C24.304C23 C23.395C24 C23.396C24 C23.397C24 C23.406C24 C23.407C24 C23.408C24 C23.409C24 C23.411C24 C23.412C24 C23.413C24 C23.414C24 C24.309C23 C23.416C24 C23.419C24 C23.420C24 C24.311C23 C23.422C24 C23.424C24 C23.425C24 C23.428C24 C23.429C24 C23.432C24 C23.433C24 C42⋊19D4 C42⋊20D4 C42.167D4 C42⋊21D4 C42.168D4 C42.169D4 C42.170D4 C42.171D4 C42⋊6Q8 C42⋊7Q8 C42.35Q8 C24.327C23 C23.456C24 C23.458C24 C24.332C23 C42.172D4 C42.173D4 C42.174D4 C42.175D4 C42.176D4 C42.177D4 C42.36Q8 C42.37Q8 C23.473C24 C24.338C23 C24.339C23 C24.341C23 C42.181D4 C23.485C24 C23.486C24 C24.345C23 C23.488C24 C24.346C23 C23.490C24 C42.182D4 C23.493C24 C23.494C24 C24.347C23 C23.496C24 C24.348C23 C42⋊23D4 C42⋊8Q8 C42.38Q8 C23.550C24 C23.551C24 C23.554C24 C23.555C24 C42⋊32D4 C42.198D4 C42⋊11Q8
C2p.(C4×D4): D4⋊C42 Q8⋊C42 C4⋊C8⋊13C4 C4⋊C8⋊14C4 D4⋊C4⋊C4 C4.67(C4×D4) C4.68(C4×D4) C2.(C4×Q16) ...
C4×C4⋊C4 is a maximal quotient of
C24.624C23 C24.625C23 C24.626C23 C8⋊C42
C2p.(C4×D4): C43.7C2 C42.45Q8 C4⋊C8⋊13C4 C4⋊C8⋊14C4 C8.14C42 C8.5C42 C8.6C42 Dic3⋊C42 ...
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 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 |
| kernel | C4×C4⋊C4 | C2.C42 | C2×C42 | C2×C4⋊C4 | C42 | C4⋊C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 3 | 2 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C4×C4⋊C4 ►in GL4(𝔽5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 3 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,3,0,0,1,1],[4,0,0,0,0,2,0,0,0,0,3,0,0,0,2,2] >;
C4×C4⋊C4 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes C_4
% in TeX
G:=Group("C4xC4:C4"); // GroupNames label
G:=SmallGroup(64,59);
// by ID
G=gap.SmallGroup(64,59);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,122]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations