Copied to
clipboard

G = C3×C62⋊5C4order 432 = 24·33

Direct product of C3 and C62⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C62⋊5C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — C6×C3⋊Dic3 — C3×C62⋊5C4
 Lower central C32 — C3×C6 — C3×C62⋊5C4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C3×C625C4
G = < a,b,c,d | a3=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c3, dcd-1=c-1 >

Subgroups: 756 in 332 conjugacy classes, 102 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3⋊Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C32×C6, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C2×C62, C3×C3⋊Dic3, C3×C62, C3×C62, C3×C62, C3×C6.D4, C625C4, C6×C3⋊Dic3, C63, C3×C625C4
Quotients:

Smallest permutation representation of C3×C625C4
On 72 points
Generators in S72
(1 14 12)(2 15 10)(3 13 11)(4 22 20)(5 23 21)(6 24 19)(7 33 26)(8 31 27)(9 32 25)(16 30 34)(17 28 35)(18 29 36)(37 53 46)(38 54 47)(39 49 48)(40 50 43)(41 51 44)(42 52 45)(55 69 62)(56 70 63)(57 71 64)(58 72 65)(59 67 66)(60 68 61)
(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 36 13 17 10 30)(2 34 14 18 11 28)(3 35 15 16 12 29)(4 8 24 33 21 25)(5 9 22 31 19 26)(6 7 23 32 20 27)(37 54 48 40 51 45)(38 49 43 41 52 46)(39 50 44 42 53 47)(55 70 64 58 67 61)(56 71 65 59 68 62)(57 72 66 60 69 63)
(1 38 31 62)(2 40 32 64)(3 42 33 66)(4 63 16 39)(5 65 17 41)(6 61 18 37)(7 67 11 45)(8 69 12 47)(9 71 10 43)(13 52 26 59)(14 54 27 55)(15 50 25 57)(19 68 36 46)(20 70 34 48)(21 72 35 44)(22 56 30 49)(23 58 28 51)(24 60 29 53)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A ··· 6F 6G ··· 6CP 12A ··· 12H order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 1 1 2 ··· 2 18 18 18 18 1 ··· 1 2 ··· 2 18 ··· 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 S3×C6 C3×C3⋊D4 kernel C3×C62⋊5C4 C6×C3⋊Dic3 C63 C62⋊5C4 C3×C62 C2×C3⋊Dic3 C2×C62 C62 C2×C62 C32×C6 C62 C62 C22×C6 C3×C6 C3×C6 C2×C6 C2×C6 C6 # reps 1 2 1 2 4 4 2 8 4 2 8 4 8 16 4 16 8 32

Matrix representation of C3×C625C4 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 9 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 10 0 0 0 0 0 0 4
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C625C4 in GAP, Magma, Sage, TeX

C_3\times C_6^2\rtimes_5C_4
% in TeX

G:=Group("C3xC6^2:5C4");
// GroupNames label

G:=SmallGroup(432,495);
// by ID

G=gap.SmallGroup(432,495);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^3,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽