Copied to
clipboard

G = C2×C62order 72 = 23·32

Abelian group of type [2,6,6]

direct product, abelian, monomial

Aliases: C2×C62, SmallGroup(72,50)

Series: Derived Chief Lower central Upper central

C1 — C2×C62
C1C3C32C3×C6C62 — C2×C62
C1 — C2×C62
C1 — C2×C62

Generators and relations for C2×C62
 G = < a,b,c | a2=b6=c6=1, ab=ba, ac=ca, bc=cb >

Subgroups: 96, all normal (4 characteristic)
C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], C23, C32, C2×C6 [×28], C3×C6 [×7], C22×C6 [×4], C62 [×7], C2×C62
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], C23, C32, C2×C6 [×28], C3×C6 [×7], C22×C6 [×4], C62 [×7], C2×C62

Smallest permutation representation of C2×C62
Regular action on 72 points
Generators in S72
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 51)(14 52)(15 53)(16 54)(17 49)(18 50)(19 63)(20 64)(21 65)(22 66)(23 61)(24 62)(25 59)(26 60)(27 55)(28 56)(29 57)(30 58)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(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 65 66)(67 68 69 70 71 72)
(1 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 45 67 29 49)(8 23 46 68 30 50)(9 24 47 69 25 51)(10 19 48 70 26 52)(11 20 43 71 27 53)(12 21 44 72 28 54)

G:=sub<Sym(72)| (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,59)(26,60)(27,55)(28,56)(29,57)(30,58)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (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)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,45,67,29,49)(8,23,46,68,30,50)(9,24,47,69,25,51)(10,19,48,70,26,52)(11,20,43,71,27,53)(12,21,44,72,28,54)>;

G:=Group( (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,59)(26,60)(27,55)(28,56)(29,57)(30,58)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (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)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,45,67,29,49)(8,23,46,68,30,50)(9,24,47,69,25,51)(10,19,48,70,26,52)(11,20,43,71,27,53)(12,21,44,72,28,54) );

G=PermutationGroup([(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,51),(14,52),(15,53),(16,54),(17,49),(18,50),(19,63),(20,64),(21,65),(22,66),(23,61),(24,62),(25,59),(26,60),(27,55),(28,56),(29,57),(30,58),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)], [(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),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,32,58,18,39,61),(2,33,59,13,40,62),(3,34,60,14,41,63),(4,35,55,15,42,64),(5,36,56,16,37,65),(6,31,57,17,38,66),(7,22,45,67,29,49),(8,23,46,68,30,50),(9,24,47,69,25,51),(10,19,48,70,26,52),(11,20,43,71,27,53),(12,21,44,72,28,54)])

C2×C62 is a maximal subgroup of   C625C4

72 conjugacy classes

class 1 2A···2G3A···3H6A···6BD
order12···23···36···6
size11···11···11···1

72 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC2×C62C62C22×C6C2×C6
# reps17856

Matrix representation of C2×C62 in GL3(𝔽7) generated by

100
060
001
,
100
020
006
,
500
050
004
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,1],[1,0,0,0,2,0,0,0,6],[5,0,0,0,5,0,0,0,4] >;

C2×C62 in GAP, Magma, Sage, TeX

C_2\times C_6^2
% in TeX

G:=Group("C2xC6^2");
// GroupNames label

G:=SmallGroup(72,50);
// by ID

G=gap.SmallGroup(72,50);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽