Copied to
clipboard

## G = C11×2+ 1+4order 352 = 25·11

### Direct product of C11 and 2+ 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×2+ 1+4
 Chief series C1 — C2 — C22 — C2×C22 — D4×C11 — D4×C22 — C11×2+ 1+4
 Lower central C1 — C2 — C11×2+ 1+4
 Upper central C1 — C22 — C11×2+ 1+4

Generators and relations for C11×2+ 1+4
G = < a,b,c,d,e | a11=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C11, C2×D4, C4○D4, C22, C22, 2+ 1+4, C44, C2×C22, C2×C22, C2×C44, D4×C11, Q8×C11, C22×C22, D4×C22, C11×C4○D4, C11×2+ 1+4
Quotients: C1, C2, C22, C23, C11, C24, C22, 2+ 1+4, C2×C22, C22×C22, C23×C22, C11×2+ 1+4

Smallest permutation representation of C11×2+ 1+4
On 88 points
Generators in S88
(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 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)
(1 75 31 60)(2 76 32 61)(3 77 33 62)(4 67 23 63)(5 68 24 64)(6 69 25 65)(7 70 26 66)(8 71 27 56)(9 72 28 57)(10 73 29 58)(11 74 30 59)(12 41 85 45)(13 42 86 46)(14 43 87 47)(15 44 88 48)(16 34 78 49)(17 35 79 50)(18 36 80 51)(19 37 81 52)(20 38 82 53)(21 39 83 54)(22 40 84 55)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 34)(9 35)(10 36)(11 37)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(1 60 31 75)(2 61 32 76)(3 62 33 77)(4 63 23 67)(5 64 24 68)(6 65 25 69)(7 66 26 70)(8 56 27 71)(9 57 28 72)(10 58 29 73)(11 59 30 74)(12 41 85 45)(13 42 86 46)(14 43 87 47)(15 44 88 48)(16 34 78 49)(17 35 79 50)(18 36 80 51)(19 37 81 52)(20 38 82 53)(21 39 83 54)(22 40 84 55)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 34)(9 35)(10 36)(11 37)(12 63)(13 64)(14 65)(15 66)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(67 85)(68 86)(69 87)(70 88)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)

G:=sub<Sym(88)| (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)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,75,31,60)(2,76,32,61)(3,77,33,62)(4,67,23,63)(5,68,24,64)(6,69,25,65)(7,70,26,66)(8,71,27,56)(9,72,28,57)(10,73,29,58)(11,74,30,59)(12,41,85,45)(13,42,86,46)(14,43,87,47)(15,44,88,48)(16,34,78,49)(17,35,79,50)(18,36,80,51)(19,37,81,52)(20,38,82,53)(21,39,83,54)(22,40,84,55), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,34)(9,35)(10,36)(11,37)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (1,60,31,75)(2,61,32,76)(3,62,33,77)(4,63,23,67)(5,64,24,68)(6,65,25,69)(7,66,26,70)(8,56,27,71)(9,57,28,72)(10,58,29,73)(11,59,30,74)(12,41,85,45)(13,42,86,46)(14,43,87,47)(15,44,88,48)(16,34,78,49)(17,35,79,50)(18,36,80,51)(19,37,81,52)(20,38,82,53)(21,39,83,54)(22,40,84,55), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,34)(9,35)(10,36)(11,37)(12,63)(13,64)(14,65)(15,66)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(67,85)(68,86)(69,87)(70,88)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)>;

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)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,75,31,60)(2,76,32,61)(3,77,33,62)(4,67,23,63)(5,68,24,64)(6,69,25,65)(7,70,26,66)(8,71,27,56)(9,72,28,57)(10,73,29,58)(11,74,30,59)(12,41,85,45)(13,42,86,46)(14,43,87,47)(15,44,88,48)(16,34,78,49)(17,35,79,50)(18,36,80,51)(19,37,81,52)(20,38,82,53)(21,39,83,54)(22,40,84,55), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,34)(9,35)(10,36)(11,37)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (1,60,31,75)(2,61,32,76)(3,62,33,77)(4,63,23,67)(5,64,24,68)(6,65,25,69)(7,66,26,70)(8,56,27,71)(9,57,28,72)(10,58,29,73)(11,59,30,74)(12,41,85,45)(13,42,86,46)(14,43,87,47)(15,44,88,48)(16,34,78,49)(17,35,79,50)(18,36,80,51)(19,37,81,52)(20,38,82,53)(21,39,83,54)(22,40,84,55), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,34)(9,35)(10,36)(11,37)(12,63)(13,64)(14,65)(15,66)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(67,85)(68,86)(69,87)(70,88)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84) );

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),(56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88)], [(1,75,31,60),(2,76,32,61),(3,77,33,62),(4,67,23,63),(5,68,24,64),(6,69,25,65),(7,70,26,66),(8,71,27,56),(9,72,28,57),(10,73,29,58),(11,74,30,59),(12,41,85,45),(13,42,86,46),(14,43,87,47),(15,44,88,48),(16,34,78,49),(17,35,79,50),(18,36,80,51),(19,37,81,52),(20,38,82,53),(21,39,83,54),(22,40,84,55)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,34),(9,35),(10,36),(11,37),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(1,60,31,75),(2,61,32,76),(3,62,33,77),(4,63,23,67),(5,64,24,68),(6,65,25,69),(7,66,26,70),(8,56,27,71),(9,57,28,72),(10,58,29,73),(11,59,30,74),(12,41,85,45),(13,42,86,46),(14,43,87,47),(15,44,88,48),(16,34,78,49),(17,35,79,50),(18,36,80,51),(19,37,81,52),(20,38,82,53),(21,39,83,54),(22,40,84,55)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,34),(9,35),(10,36),(11,37),(12,63),(13,64),(14,65),(15,66),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(67,85),(68,86),(69,87),(70,88),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)]])

187 conjugacy classes

 class 1 2A 2B ··· 2J 4A ··· 4F 11A ··· 11J 22A ··· 22J 22K ··· 22CV 44A ··· 44BH order 1 2 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

187 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C11 C22 C22 2+ 1+4 C11×2+ 1+4 kernel C11×2+ 1+4 D4×C22 C11×C4○D4 2+ 1+4 C2×D4 C4○D4 C11 C1 # reps 1 9 6 10 90 60 1 10

Matrix representation of C11×2+ 1+4 in GL4(𝔽89) generated by

 39 0 0 0 0 39 0 0 0 0 39 0 0 0 0 39
,
 59 72 39 78 0 0 0 88 2 26 30 55 0 1 0 0
,
 26 26 51 77 87 63 59 34 0 0 0 1 0 0 1 0
,
 30 51 50 37 0 0 0 88 87 63 59 34 0 1 0 0
,
 26 26 17 18 87 63 59 34 0 0 0 88 0 0 88 0
G:=sub<GL(4,GF(89))| [39,0,0,0,0,39,0,0,0,0,39,0,0,0,0,39],[59,0,2,0,72,0,26,1,39,0,30,0,78,88,55,0],[26,87,0,0,26,63,0,0,51,59,0,1,77,34,1,0],[30,0,87,0,51,0,63,1,50,0,59,0,37,88,34,0],[26,87,0,0,26,63,0,0,17,59,0,88,18,34,88,0] >;

C11×2+ 1+4 in GAP, Magma, Sage, TeX

C_{11}\times 2_+^{1+4}
% in TeX

G:=Group("C11xES+(2,2)");
// GroupNames label

G:=SmallGroup(352,192);
// by ID

G=gap.SmallGroup(352,192);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-11,-2,2137,1628,4419]);
// Polycyclic

G:=Group<a,b,c,d,e|a^11=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations

׿
×
𝔽