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G = C22×C42order 64 = 26

Abelian group of type [2,2,4,4]

direct product, p-group, abelian, monomial

Aliases: C22×C42, SmallGroup(64,192)

Series: Derived Chief Lower central Upper central Jennings

C1 — C22×C42
C1C2C22C23C24C23×C4 — C22×C42
C1 — C22×C42
C1 — C22×C42
C1C22 — 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 [×15], C4 [×24], C22, C22 [×34], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], C22×C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], C22×C42

Smallest permutation representation of 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  C232C42  C23.28C42  C42.378D4  C42.425D4  C23.32M4(2)  (C2×C4)≀C2  C23.165C24  C23.167C24  C4242D4  C23.178C24  C23.179C24  C4246D4  C42.439D4  C4243D4  C23.753C24  C24.598C23  C24.599C23  C4247D4  C42.440D4  C42.677C23  C22.33C25  C422A4
C22×C42 is a maximal quotient of
C23⋊C42  C24.524C23  D44C42  Q84C42  M4(2)○2M4(2)  D4.5C42

64 conjugacy classes

class 1 2A···2O4A···4AV
order12···24···4
size11···11···1

64 irreducible representations

dim1111
type+++
imageC1C2C2C4
kernelC22×C42C2×C42C23×C4C22×C4
# reps112348

Matrix representation of C22×C42 in GL4(𝔽5) generated by

1000
0400
0010
0001
,
4000
0400
0010
0004
,
1000
0300
0020
0003
,
2000
0400
0030
0003
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

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