Copied to
clipboard

## G = C2×D4×3- 1+2order 432 = 24·33

### Direct product of C2, D4 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D4×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C22×3- 1+2 — D4×3- 1+2 — C2×D4×3- 1+2
 Lower central C1 — C6 — C2×D4×3- 1+2
 Upper central C1 — C2×C6 — C2×D4×3- 1+2

Generators and relations for C2×D4×3- 1+2
G = < a,b,c,d,e | a2=b4=c2=d9=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 350 in 216 conjugacy classes, 133 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C9, C32, C12, C12, C2×C6, C2×C6, C2×C6, C2×D4, C18, C18, C3×C6, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, 3- 1+2, C36, C2×C18, C2×C18, C3×C12, C62, C62, C62, C6×D4, C6×D4, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×C36, D4×C9, C22×C18, C6×C12, D4×C32, C2×C62, C4×3- 1+2, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, D4×C18, D4×C3×C6, C2×C4×3- 1+2, D4×3- 1+2, C23×3- 1+2, C2×D4×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, 3- 1+2, C62, C6×D4, C2×3- 1+2, D4×C32, C2×C62, C22×3- 1+2, D4×C3×C6, D4×3- 1+2, C23×3- 1+2, C2×D4×3- 1+2

Smallest permutation representation of C2×D4×3- 1+2
On 72 points
Generators in S72
(1 45)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(55 71)(56 72)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(1 18 27 70)(2 10 19 71)(3 11 20 72)(4 12 21 64)(5 13 22 65)(6 14 23 66)(7 15 24 67)(8 16 25 68)(9 17 26 69)(28 57 39 46)(29 58 40 47)(30 59 41 48)(31 60 42 49)(32 61 43 50)(33 62 44 51)(34 63 45 52)(35 55 37 53)(36 56 38 54)
(1 70)(2 71)(3 72)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 61 58)(56 59 62)(65 71 68)(66 69 72)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6F 6G ··· 6N 6O ··· 6T 6U ··· 6AB 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18R 18S ··· 18AP 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 1 1 3 3 2 2 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 3 ··· 3 2 2 2 2 6 6 6 6 3 ··· 3 6 ··· 6 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 6 type + + + + + image C1 C2 C2 C2 C3 C3 C6 C6 C6 C6 C6 C6 D4 C3×D4 C3×D4 3- 1+2 C2×3- 1+2 C2×3- 1+2 C2×3- 1+2 D4×3- 1+2 kernel C2×D4×3- 1+2 C2×C4×3- 1+2 D4×3- 1+2 C23×3- 1+2 D4×C18 D4×C3×C6 C2×C36 D4×C9 C22×C18 C6×C12 D4×C32 C2×C62 C2×3- 1+2 C18 C3×C6 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 4 2 6 2 6 24 12 2 8 4 2 12 4 2 2 8 4 4

Matrix representation of C2×D4×3- 1+2 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 1 0 0 0 35 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 0 0 0 0 2 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 10 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 26 0 0 1 0 0
,
 10 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 10

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,35,0,0,0,1,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,2,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,26,0],[10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;

C2×D4×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times D_4\times 3_-^{1+2}
% in TeX

G:=Group("C2xD4xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,405);
// by ID

G=gap.SmallGroup(432,405);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,528,760]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^9=e^3=1,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^-1=d^4>;
// generators/relations

׿
×
𝔽