Copied to
clipboard

## G = C2×C33⋊8D4order 432 = 24·33

### Direct product of C2 and C33⋊8D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C2×C33⋊8D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8D4 — C2×C33⋊8D4
 Lower central C33 — C32×C6 — C2×C33⋊8D4
 Upper central C1 — C22

Generators and relations for C2×C338D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 3160 in 452 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×D12, C2×C3⋊D4, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, C3⋊D12, C6×Dic3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C32×Dic3, C6×C3⋊S3, C6×C3⋊S3, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C2×C3⋊D12, C2×C12⋊S3, C338D4, Dic3×C3×C6, C2×C6×C3⋊S3, C22×C33⋊C2, C2×C338D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C3⋊D4, C22×S3, S32, C2×C3⋊S3, C2×D12, C2×C3⋊D4, C3⋊D12, C12⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×C3⋊D12, C2×C12⋊S3, C338D4, C2×S3×C3⋊S3, C2×C338D4

Smallest permutation representation of C2×C338D4
On 72 points
Generators in S72
(1 29)(2 30)(3 31)(4 32)(5 43)(6 44)(7 41)(8 42)(9 13)(10 14)(11 15)(12 16)(17 27)(18 28)(19 25)(20 26)(21 53)(22 54)(23 55)(24 56)(33 68)(34 65)(35 66)(36 67)(37 58)(38 59)(39 60)(40 57)(45 72)(46 69)(47 70)(48 71)(49 64)(50 61)(51 62)(52 63)
(1 71 27)(2 72 28)(3 69 25)(4 70 26)(5 38 53)(6 39 54)(7 40 55)(8 37 56)(9 36 63)(10 33 64)(11 34 61)(12 35 62)(13 67 52)(14 68 49)(15 65 50)(16 66 51)(17 29 48)(18 30 45)(19 31 46)(20 32 47)(21 43 59)(22 44 60)(23 41 57)(24 42 58)
(1 61 56)(2 62 53)(3 63 54)(4 64 55)(5 72 12)(6 69 9)(7 70 10)(8 71 11)(13 44 46)(14 41 47)(15 42 48)(16 43 45)(17 65 58)(18 66 59)(19 67 60)(20 68 57)(21 30 51)(22 31 52)(23 32 49)(24 29 50)(25 36 39)(26 33 40)(27 34 37)(28 35 38)
(1 8 34)(2 35 5)(3 6 36)(4 33 7)(9 25 54)(10 55 26)(11 27 56)(12 53 28)(13 19 22)(14 23 20)(15 17 24)(16 21 18)(29 42 65)(30 66 43)(31 44 67)(32 68 41)(37 61 71)(38 72 62)(39 63 69)(40 70 64)(45 51 59)(46 60 52)(47 49 57)(48 58 50)
(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 29)(2 32)(3 31)(4 30)(5 68)(6 67)(7 66)(8 65)(9 60)(10 59)(11 58)(12 57)(13 39)(14 38)(15 37)(16 40)(17 71)(18 70)(19 69)(20 72)(21 64)(22 63)(23 62)(24 61)(25 46)(26 45)(27 48)(28 47)(33 43)(34 42)(35 41)(36 44)(49 53)(50 56)(51 55)(52 54)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6O 6P ··· 6AA 6AB 6AC 6AD 6AE 12A ··· 12P order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 1 1 18 18 54 54 2 ··· 2 4 4 4 4 6 6 2 ··· 2 4 ··· 4 18 18 18 18 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D12 C3⋊D4 S32 C3⋊D12 C2×S32 kernel C2×C33⋊8D4 C33⋊8D4 Dic3×C3×C6 C2×C6×C3⋊S3 C22×C33⋊C2 C6×Dic3 C22×C3⋊S3 C32×C6 C3×Dic3 C2×C3⋊S3 C62 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 4 1 1 1 4 1 2 8 2 5 16 4 4 8 4

Matrix representation of C2×C338D4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0
,
 4 6 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 3 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[4,8,0,0,0,0,0,0,6,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;

C2×C338D4 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_8D_4
% in TeX

G:=Group("C2xC3^3:8D4");
// GroupNames label

G:=SmallGroup(432,682);
// by ID

G=gap.SmallGroup(432,682);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,571,2028,14118]);
// Polycyclic

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

׿
×
𝔽