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## G = C2×D36⋊C3order 432 = 24·33

### Direct product of C2 and D36⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×D36⋊C3
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — C2×D36⋊C3
 Lower central C9 — C18 — C2×D36⋊C3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D36⋊C3
G = < a,b,c,d | a2=b36=c2=d3=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b13, dcd-1=b12c >

Subgroups: 782 in 170 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, 3- 1+2, C36, C36, D18, D18, C2×C18, C2×C18, C3×C12, S3×C6, C62, C2×D12, C6×D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, D36, C2×C36, C2×C36, C22×D9, C3×D12, C6×C12, S3×C2×C6, C4×3- 1+2, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, C2×D36, C6×D12, D36⋊C3, C2×C4×3- 1+2, C22×C9⋊C6, C2×D36⋊C3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, S3×C6, C2×D12, C6×D4, C9⋊C6, C3×D12, S3×C2×C6, C2×C9⋊C6, C6×D12, D36⋊C3, C22×C9⋊C6, C2×D36⋊C3

Smallest permutation representation of C2×D36⋊C3
On 72 points
Generators in S72
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)
(2 26 14)(3 15 27)(5 29 17)(6 18 30)(8 32 20)(9 21 33)(11 35 23)(12 24 36)(37 61 49)(38 50 62)(40 64 52)(41 53 65)(43 67 55)(44 56 68)(46 70 58)(47 59 71)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D ··· 6I 6J ··· 6Q 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18I 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 ··· 6 6 ··· 6 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 18 18 2 3 3 2 2 2 2 2 3 ··· 3 18 ··· 18 6 6 6 2 2 2 2 6 6 6 6 6 ··· 6 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 C3×D4 D12 S3×C6 S3×C6 C3×D12 C9⋊C6 C2×C9⋊C6 C2×C9⋊C6 D36⋊C3 kernel C2×D36⋊C3 D36⋊C3 C2×C4×3- 1+2 C22×C9⋊C6 C2×D36 D36 C2×C36 C22×D9 C6×C12 C2×3- 1+2 C3×C12 C62 C2×C12 C18 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 1 2 2 1 2 4 4 4 2 8 1 2 1 4

Matrix representation of C2×D36⋊C3 in GL8(𝔽37)

 36 0 0 0 0 0 0 0 0 36 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 1
,
 0 36 0 0 0 0 0 0 1 36 0 0 0 0 0 0 0 0 0 0 0 0 10 5 0 0 0 0 0 0 32 5 0 0 32 27 0 0 0 0 0 0 10 5 0 0 0 0 0 0 0 0 32 27 0 0 0 0 0 0 10 5 0 0
,
 36 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0
,
 10 0 0 0 0 0 0 0 0 10 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 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36

G:=sub<GL(8,GF(37))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,0,32,10,0,0,0,0,0,0,27,5,0,0,0,0,0,0,0,0,32,10,0,0,0,0,0,0,27,5,0,0,10,32,0,0,0,0,0,0,5,5,0,0,0,0],[36,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36] >;

C2×D36⋊C3 in GAP, Magma, Sage, TeX

C_2\times D_{36}\rtimes C_3
% in TeX

G:=Group("C2xD36:C3");
// GroupNames label

G:=SmallGroup(432,354);
// by ID

G=gap.SmallGroup(432,354);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,10085,1034,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^36=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^12*c>;
// generators/relations

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