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## G = C3×D36⋊5C2order 432 = 24·33

### Direct product of C3 and D36⋊5C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×D36⋊5C2
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C12×D9 — C3×D36⋊5C2
 Lower central C9 — C18 — C3×D36⋊5C2
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×D365C2
G = < a,b,c,d | a3=b36=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b18c >

Subgroups: 470 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, C9, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C4○D4, D9 [×2], C18, C18 [×4], C3×S3 [×2], C3×C6, C3×C6, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C3×C9, Dic9 [×2], C36 [×2], C36 [×2], D18 [×2], C2×C18, C2×C18, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, C4○D12, C3×C4○D4, C3×D9 [×2], C3×C18, C3×C18, Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, C2×C36, C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C3×Dic9 [×2], C3×C36 [×2], C6×D9 [×2], C6×C18, D365C2, C3×C4○D12, C3×Dic18, C12×D9 [×2], C3×D36, C3×C9⋊D4 [×2], C6×C36, C3×D365C2
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4, D9, C3×S3, C22×S3, C22×C6, D18 [×3], S3×C6 [×3], C4○D12, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9 [×3], D365C2, C3×C4○D12, C2×C6×D9, C3×D365C2

Smallest permutation representation of C3×D365C2
On 72 points
Generators in S72
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 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)
(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 60)(36 59)
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6M 6N 6O 6P 6Q 9A ··· 9I 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 18A ··· 18AA 36A ··· 36AJ order 1 2 2 2 2 3 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 18 ··· 18 36 ··· 36 size 1 1 2 18 18 1 1 2 2 2 1 1 2 18 18 1 1 2 ··· 2 18 18 18 18 2 ··· 2 1 1 1 1 2 ··· 2 18 18 18 18 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 D9 C3×S3 D18 S3×C6 D18 S3×C6 C3×C4○D4 C4○D12 C3×D9 C6×D9 C6×D9 D36⋊5C2 C3×C4○D12 C3×D36⋊5C2 kernel C3×D36⋊5C2 C3×Dic18 C12×D9 C3×D36 C3×C9⋊D4 C6×C36 D36⋊5C2 Dic18 C4×D9 D36 C9⋊D4 C2×C36 C6×C12 C3×C12 C62 C3×C9 C2×C12 C2×C12 C12 C12 C2×C6 C2×C6 C9 C32 C2×C4 C4 C22 C3 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 2 1 2 3 2 6 4 3 2 4 4 6 12 6 12 8 24

Matrix representation of C3×D365C2 in GL2(𝔽37) generated by

 10 0 0 10
,
 22 0 0 32
,
 0 32 22 0
,
 1 0 0 36
G:=sub<GL(2,GF(37))| [10,0,0,10],[22,0,0,32],[0,22,32,0],[1,0,0,36] >;

C3×D365C2 in GAP, Magma, Sage, TeX

C_3\times D_{36}\rtimes_5C_2
% in TeX

G:=Group("C3xD36:5C2");
// GroupNames label

G:=SmallGroup(432,344);
// by ID

G=gap.SmallGroup(432,344);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^36=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^18*c>;
// generators/relations

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