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## G = C2×Dic9⋊C6order 432 = 24·33

### Direct product of C2 and Dic9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×Dic9⋊C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — C2×Dic9⋊C6
 Lower central C9 — C18 — C2×Dic9⋊C6
 Upper central C1 — C22 — C23

Generators and relations for C2×Dic9⋊C6
G = < a,b,c,d | a2=b18=d6=1, c2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, dcd-1=b9c >

Subgroups: 654 in 178 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, 3- 1+2, Dic9, D18, D18, C2×C18, C2×C18, C2×C18, C3×Dic3, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C22×C18, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C9⋊C12, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, C2×C9⋊D4, C6×C3⋊D4, C2×C9⋊C12, Dic9⋊C6, C22×C9⋊C6, C23×3- 1+2, C2×Dic9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊D4, C6×D4, C9⋊C6, C3×C3⋊D4, S3×C2×C6, C2×C9⋊C6, C6×C3⋊D4, Dic9⋊C6, C22×C9⋊C6, C2×Dic9⋊C6

Smallest permutation representation of C2×Dic9⋊C6
On 72 points
Generators in S72
(1 34)(2 35)(3 36)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(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)
(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 46 10 37)(2 45 11 54)(3 44 12 53)(4 43 13 52)(5 42 14 51)(6 41 15 50)(7 40 16 49)(8 39 17 48)(9 38 18 47)(19 61 28 70)(20 60 29 69)(21 59 30 68)(22 58 31 67)(23 57 32 66)(24 56 33 65)(25 55 34 64)(26 72 35 63)(27 71 36 62)
(1 34)(2 29 8 35 14 23)(3 24 15 36 9 30)(4 19)(5 32 11 20 17 26)(6 27 18 21 12 33)(7 22)(10 25)(13 28)(16 31)(37 64)(38 59 44 65 50 71)(39 72 51 66 45 60)(40 67)(41 62 47 68 53 56)(42 57 54 69 48 63)(43 70)(46 55)(49 58)(52 61)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A ··· 6G 6H ··· 6M 6N 6O 6P 6Q 6R 6S 6T 6U 9A 9B 9C 12A 12B 12C 12D 18A ··· 18U order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 6 6 6 9 9 9 12 12 12 12 18 ··· 18 size 1 1 1 1 2 2 18 18 2 3 3 18 18 2 ··· 2 3 ··· 3 6 6 6 6 18 18 18 18 6 6 6 18 18 18 18 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C3×S3 C3×D4 C3⋊D4 S3×C6 C3×C3⋊D4 C9⋊C6 C2×C9⋊C6 Dic9⋊C6 kernel C2×Dic9⋊C6 C2×C9⋊C12 Dic9⋊C6 C22×C9⋊C6 C23×3- 1+2 C2×C9⋊D4 C2×Dic9 C9⋊D4 C22×D9 C22×C18 C2×C62 C2×3- 1+2 C62 C22×C6 C18 C3×C6 C2×C6 C6 C23 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 2 3 2 4 4 6 8 1 3 4

Matrix representation of C2×Dic9⋊C6 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
,
 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 11 9 0 0 0 0 0 0 0 26 27 0 0 0 0 0 26 1 0 0 0 0 0 0 10 18 18 26 1 9 0 0 13 0 0 11 0 0 0 0 0 32 5 10 26 11
,
 25 3 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 34 0 0 35 0 0 0 0 0 0 33 23 31 0 0 0 2 30 33 31 8 35 0 0 5 0 0 3 0 0 0 0 24 11 15 34 7 26 0 0 0 21 27 1 33 34
,
 27 0 0 0 0 0 0 0 31 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 26 0 0 0 0 0 0 36 0 10 0 0 0 0 0 34 0 0 36 0 0 0 0 15 0 13 0 11 0 0 0 30 22 12 11 10 27

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],[36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,11,0,26,10,13,0,0,0,9,26,1,18,0,32,0,0,0,27,0,18,0,5,0,0,0,0,0,26,11,10,0,0,0,0,0,1,0,26,0,0,0,0,0,9,0,11],[25,1,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,34,0,2,5,24,0,0,0,0,0,30,0,11,21,0,0,0,33,33,0,15,27,0,0,35,23,31,3,34,1,0,0,0,31,8,0,7,33,0,0,0,0,35,0,26,34],[27,31,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,10,36,34,15,30,0,0,0,26,0,0,0,22,0,0,0,0,10,0,13,12,0,0,0,0,0,36,0,11,0,0,0,0,0,0,11,10,0,0,0,0,0,0,0,27] >;

C2×Dic9⋊C6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_9\rtimes C_6
% in TeX

G:=Group("C2xDic9:C6");
// GroupNames label

G:=SmallGroup(432,379);
// by ID

G=gap.SmallGroup(432,379);
# by ID

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

G:=Group<a,b,c,d|a^2=b^18=d^6=1,c^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^9*c>;
// generators/relations

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