direct product, p-group, abelian, monomial
Aliases: C22×C42, SmallGroup(64,192)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C22×C42 |
| C1 — C22×C42 |
| C1 — C22×C42 |
Generators and relations for C22×C42
G = < a,b,c,d | a2=b2=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 249, all normal (4 characteristic)
C1, C2, C4, C22, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, C22×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, C22×C42
►Regular action on 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(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 59 51 15)(2 60 52 16)(3 57 49 13)(4 58 50 14)(5 41 36 25)(6 42 33 26)(7 43 34 27)(8 44 35 28)(9 61 53 17)(10 62 54 18)(11 63 55 19)(12 64 56 20)(21 45 37 29)(22 46 38 30)(23 47 39 31)(24 48 40 32)
G:=sub<Sym(64)| (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,39)(2,40)(3,37)(4,38)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (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), (1,59,51,15)(2,60,52,16)(3,57,49,13)(4,58,50,14)(5,41,36,25)(6,42,33,26)(7,43,34,27)(8,44,35,28)(9,61,53,17)(10,62,54,18)(11,63,55,19)(12,64,56,20)(21,45,37,29)(22,46,38,30)(23,47,39,31)(24,48,40,32)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,45)(6,46)(7,47)(8,48)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,39)(2,40)(3,37)(4,38)(5,17)(6,18)(7,19)(8,20)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,62)(34,63)(35,64)(36,61), (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), (1,59,51,15)(2,60,52,16)(3,57,49,13)(4,58,50,14)(5,41,36,25)(6,42,33,26)(7,43,34,27)(8,44,35,28)(9,61,53,17)(10,62,54,18)(11,63,55,19)(12,64,56,20)(21,45,37,29)(22,46,38,30)(23,47,39,31)(24,48,40,32) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,45),(6,46),(7,47),(8,48),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28),(29,36),(30,33),(31,34),(32,35),(37,41),(38,42),(39,43),(40,44),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,39),(2,40),(3,37),(4,38),(5,17),(6,18),(7,19),(8,20),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,62),(34,63),(35,64),(36,61)], [(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)], [(1,59,51,15),(2,60,52,16),(3,57,49,13),(4,58,50,14),(5,41,36,25),(6,42,33,26),(7,43,34,27),(8,44,35,28),(9,61,53,17),(10,62,54,18),(11,63,55,19),(12,64,56,20),(21,45,37,29),(22,46,38,30),(23,47,39,31),(24,48,40,32)]])
C22×C42 is a maximal subgroup of
C24.624C23 C24.625C23 C24.626C23 C23⋊2C42 C23.28C42 C42.378D4 C42.425D4 C23.32M4(2) (C2×C4)≀C2 C23.165C24 C23.167C24 C42⋊42D4 C23.178C24 C23.179C24 C42⋊46D4 C42.439D4 C42⋊43D4 C23.753C24 C24.598C23 C24.599C23 C42⋊47D4 C42.440D4 C42.677C23 C22.33C25 C42⋊2A4
C22×C42 is a maximal quotient of
C23⋊C42 C24.524C23 D4⋊4C42 Q8⋊4C42 M4(2)○2M4(2) D4.5C42
64 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | + | |
| image | C1 | C2 | C2 | C4 |
| kernel | C22×C42 | C2×C42 | C23×C4 | C22×C4 |
| # reps | 1 | 12 | 3 | 48 |
Matrix representation of C22×C42 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[2,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3] >;
C22×C42 in GAP, Magma, Sage, TeX
C_2^2\times C_4^2
% in TeX
G:=Group("C2^2xC4^2"); // GroupNames label
G:=SmallGroup(64,192);
// by ID
G=gap.SmallGroup(64,192);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,199]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations