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## G = C22×C42order 64 = 26

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

Aliases: C22×C42, SmallGroup(64,192)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C42
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42
 Lower central C1 — C22×C42
 Upper central C1 — C22×C42
 Jennings C1 — C22 — 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)])

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

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