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## 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 [×2], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×14], C6 [×38], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×14], C2×C6 [×38], C22⋊C4, C3×C6, C3×C6 [×14], C3×C6 [×38], C2×Dic3 [×8], C2×C12 [×2], C22×C6, C22×C6 [×4], C22×C6 [×4], C33, C3×Dic3 [×8], C3⋊Dic3 [×2], C62, C62 [×14], C62 [×38], C6.D4 [×4], C3×C22⋊C4, C32×C6, C32×C6 [×2], C32×C6 [×2], C6×Dic3 [×8], C2×C3⋊Dic3 [×2], C2×C62, C2×C62 [×4], C2×C62 [×4], C3×C3⋊Dic3 [×2], C3×C62, C3×C62 [×2], C3×C62 [×2], C3×C6.D4 [×4], C625C4, C6×C3⋊Dic3 [×2], C63, C3×C625C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×4], C6 [×3], C2×C4, D4 [×2], Dic3 [×8], C12 [×2], D6 [×4], C2×C6, C22⋊C4, C3×S3 [×4], C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C3×Dic3 [×8], C3⋊Dic3 [×2], S3×C6 [×4], C2×C3⋊S3, C6.D4 [×4], C3×C22⋊C4, C3×C3⋊S3, C6×Dic3 [×4], C3×C3⋊D4 [×8], C2×C3⋊Dic3, C327D4 [×2], C3×C3⋊Dic3 [×2], C6×C3⋊S3, C3×C6.D4 [×4], C625C4, C6×C3⋊Dic3, C3×C327D4 [×2], C3×C625C4

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

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

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

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

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

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