Copied to
clipboard

## G = C4○D4×3- 1+2order 432 = 24·33

### Direct product of C4○D4 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4○D4×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C22×3- 1+2 — D4×3- 1+2 — C4○D4×3- 1+2
 Lower central C1 — C6 — C4○D4×3- 1+2
 Upper central C1 — C12 — C4○D4×3- 1+2

Generators and relations for C4○D4×3- 1+2
G = < a,b,c,d,e | a4=c2=d9=e3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 230 in 160 conjugacy classes, 119 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C32, C12, C12, C12, C2×C6, C2×C6, C4○D4, C18, C18, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 3- 1+2, C36, C2×C18, C3×C12, C3×C12, C62, C3×C4○D4, C3×C4○D4, C2×3- 1+2, C2×3- 1+2, C2×C36, D4×C9, Q8×C9, C6×C12, D4×C32, Q8×C32, C4×3- 1+2, C4×3- 1+2, C22×3- 1+2, C9×C4○D4, C32×C4○D4, C2×C4×3- 1+2, D4×3- 1+2, Q8×3- 1+2, C4○D4×3- 1+2
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, 3- 1+2, C62, C3×C4○D4, C2×3- 1+2, C2×C62, C22×3- 1+2, C32×C4○D4, C23×3- 1+2, C4○D4×3- 1+2

Smallest permutation representation of C4○D4×3- 1+2
On 72 points
Generators in S72
(1 45 27 34)(2 37 19 35)(3 38 20 36)(4 39 21 28)(5 40 22 29)(6 41 23 30)(7 42 24 31)(8 43 25 32)(9 44 26 33)(10 55 71 53)(11 56 72 54)(12 57 64 46)(13 58 65 47)(14 59 66 48)(15 60 67 49)(16 61 68 50)(17 62 69 51)(18 63 70 52)
(1 63 27 52)(2 55 19 53)(3 56 20 54)(4 57 21 46)(5 58 22 47)(6 59 23 48)(7 60 24 49)(8 61 25 50)(9 62 26 51)(10 37 71 35)(11 38 72 36)(12 39 64 28)(13 40 65 29)(14 41 66 30)(15 42 67 31)(16 43 68 32)(17 44 69 33)(18 45 70 34)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(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)
(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)| (1,45,27,34)(2,37,19,35)(3,38,20,36)(4,39,21,28)(5,40,22,29)(6,41,23,30)(7,42,24,31)(8,43,25,32)(9,44,26,33)(10,55,71,53)(11,56,72,54)(12,57,64,46)(13,58,65,47)(14,59,66,48)(15,60,67,49)(16,61,68,50)(17,62,69,51)(18,63,70,52), (1,63,27,52)(2,55,19,53)(3,56,20,54)(4,57,21,46)(5,58,22,47)(6,59,23,48)(7,60,24,49)(8,61,25,50)(9,62,26,51)(10,37,71,35)(11,38,72,36)(12,39,64,28)(13,40,65,29)(14,41,66,30)(15,42,67,31)(16,43,68,32)(17,44,69,33)(18,45,70,34), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(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), (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( (1,45,27,34)(2,37,19,35)(3,38,20,36)(4,39,21,28)(5,40,22,29)(6,41,23,30)(7,42,24,31)(8,43,25,32)(9,44,26,33)(10,55,71,53)(11,56,72,54)(12,57,64,46)(13,58,65,47)(14,59,66,48)(15,60,67,49)(16,61,68,50)(17,62,69,51)(18,63,70,52), (1,63,27,52)(2,55,19,53)(3,56,20,54)(4,57,21,46)(5,58,22,47)(6,59,23,48)(7,60,24,49)(8,61,25,50)(9,62,26,51)(10,37,71,35)(11,38,72,36)(12,39,64,28)(13,40,65,29)(14,41,66,30)(15,42,67,31)(16,43,68,32)(17,44,69,33)(18,45,70,34), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(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), (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([[(1,45,27,34),(2,37,19,35),(3,38,20,36),(4,39,21,28),(5,40,22,29),(6,41,23,30),(7,42,24,31),(8,43,25,32),(9,44,26,33),(10,55,71,53),(11,56,72,54),(12,57,64,46),(13,58,65,47),(14,59,66,48),(15,60,67,49),(16,61,68,50),(17,62,69,51),(18,63,70,52)], [(1,63,27,52),(2,55,19,53),(3,56,20,54),(4,57,21,46),(5,58,22,47),(6,59,23,48),(7,60,24,49),(8,61,25,50),(9,62,26,51),(10,37,71,35),(11,38,72,36),(12,39,64,28),(13,40,65,29),(14,41,66,30),(15,42,67,31),(16,43,68,32),(17,44,69,33),(18,45,70,34)], [(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(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)], [(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 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 6I 6J 6K ··· 6P 9A ··· 9F 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O ··· 12T 18A ··· 18F 18G ··· 18X 36A ··· 36L 36M ··· 36AD order 1 2 2 2 2 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 ··· 6 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 2 2 2 1 1 3 3 1 1 2 2 2 1 1 2 ··· 2 3 3 6 ··· 6 3 ··· 3 1 1 1 1 2 ··· 2 3 3 3 3 6 ··· 6 3 ··· 3 6 ··· 6 3 ··· 3 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 C4○D4 C3×C4○D4 C3×C4○D4 3- 1+2 C2×3- 1+2 C2×3- 1+2 C2×3- 1+2 C4○D4×3- 1+2 kernel C4○D4×3- 1+2 C2×C4×3- 1+2 D4×3- 1+2 Q8×3- 1+2 C9×C4○D4 C32×C4○D4 C2×C36 D4×C9 Q8×C9 C6×C12 D4×C32 Q8×C32 3- 1+2 C9 C32 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 1 6 2 18 18 6 6 6 2 2 12 4 2 6 6 2 4

Matrix representation of C4○D4×3- 1+2 in GL5(𝔽37)

 31 0 0 0 0 0 31 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 36 0 0 0 2 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 35 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 1 0 0 0 27 0 27 0 0 25 0 26
,
 26 0 0 0 0 0 26 0 0 0 0 0 1 0 11 0 0 0 10 1 0 0 0 0 26

G:=sub<GL(5,GF(37))| [31,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,2,0,0,0,36,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,35,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,11,27,25,0,0,1,0,0,0,0,0,27,26],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,11,1,26] >;

C4○D4×3- 1+2 in GAP, Magma, Sage, TeX

C_4\circ D_4\times 3_-^{1+2}
% in TeX

G:=Group("C4oD4xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,411);
// by ID

G=gap.SmallGroup(432,411);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,394,528,760]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^9=e^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,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