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## G = C2×He3⋊6D4order 432 = 24·33

### Direct product of C2 and He3⋊6D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×He3⋊6D4
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C22×C32⋊C6 — C2×He3⋊6D4
 Lower central C32 — C3×C6 — C2×He3⋊6D4
 Upper central C1 — C22 — C23

Generators and relations for C2×He36D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, ebe-1=fbf=b-1, cd=dc, ece-1=fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 969 in 221 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, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×C3⋊D4, C6×D4, C32⋊C6, C2×He3, C2×He3, C2×He3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C62, C32⋊C12, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C22×He3, C22×He3, C6×C3⋊D4, C2×C327D4, C2×C32⋊C12, He36D4, C22×C32⋊C6, C23×He3, C2×He36D4
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, C32⋊C6, C3×C3⋊D4, S3×C2×C6, C2×C32⋊C6, C6×C3⋊D4, He36D4, C22×C32⋊C6, C2×He36D4

Smallest permutation representation of C2×He36D4
On 72 points
Generators in S72
(1 40)(2 37)(3 38)(4 39)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 35)(14 36)(15 33)(16 34)(21 25)(22 26)(23 27)(24 28)(29 58)(30 59)(31 60)(32 57)(41 62)(42 63)(43 64)(44 61)(49 71)(50 72)(51 69)(52 70)(53 66)(54 67)(55 68)(56 65)
(1 18 8)(2 5 19)(3 20 6)(4 7 17)(9 45 40)(10 37 46)(11 47 38)(12 39 48)(13 49 64)(14 61 50)(15 51 62)(16 63 52)(21 55 57)(22 58 56)(23 53 59)(24 60 54)(25 68 32)(26 29 65)(27 66 30)(28 31 67)(33 69 41)(34 42 70)(35 71 43)(36 44 72)
(1 26 15)(2 16 27)(3 28 13)(4 14 25)(5 63 66)(6 67 64)(7 61 68)(8 65 62)(9 58 69)(10 70 59)(11 60 71)(12 72 57)(17 50 32)(18 29 51)(19 52 30)(20 31 49)(21 39 36)(22 33 40)(23 37 34)(24 35 38)(41 45 56)(42 53 46)(43 47 54)(44 55 48)
(1 65 29)(2 66 30)(3 67 31)(4 68 32)(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 33 45)(10 34 46)(11 35 47)(12 36 48)(21 44 72)(22 41 69)(23 42 70)(24 43 71)(25 61 50)(26 62 51)(27 63 52)(28 64 49)(37 53 59)(38 54 60)(39 55 57)(40 56 58)
(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 39)(2 38)(3 37)(4 40)(5 47)(6 46)(7 45)(8 48)(9 17)(10 20)(11 19)(12 18)(13 34)(14 33)(15 36)(16 35)(21 26)(22 25)(23 28)(24 27)(29 57)(30 60)(31 59)(32 58)(41 61)(42 64)(43 63)(44 62)(49 70)(50 69)(51 72)(52 71)(53 67)(54 66)(55 65)(56 68)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A ··· 6G 6H ··· 6M 6N ··· 6AL 6AM 6AN 6AO 6AP 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 2 2 18 18 2 3 3 6 6 6 18 18 2 ··· 2 3 ··· 3 6 ··· 6 18 18 18 18 18 18 18 18

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 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel C2×He3⋊6D4 C2×C32⋊C12 He3⋊6D4 C22×C32⋊C6 C23×He3 C2×C32⋊7D4 C2×C3⋊Dic3 C32⋊7D4 C22×C3⋊S3 C2×C62 C2×C62 C2×He3 C62 C22×C6 C3×C6 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×He36D4 in GL10(𝔽13)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12
,
 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0
,
 8 2 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 0 0 10 10 0 3 10 10 0 0 0 0 0 3 3 0 0 3 0 0 0 0 0 3 10 10 10 10 0 0 0 0 3 0 0 3 0 3 0 0 0 0 10 10 10 10 0 3 0 0 0 0 0 3 0 3 3 0
,
 5 11 0 0 0 0 0 0 0 0 12 8 0 0 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 0 5 4 0 0 0 0 0 0 0 0 0 0 10 10 0 3 10 10 0 0 0 0 0 3 3 0 0 3 0 0 0 0 0 3 10 10 10 10 0 0 0 0 3 0 0 3 0 3 0 0 0 0 10 10 10 10 0 3 0 0 0 0 0 3 0 3 3 0

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0],[8,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,0,10,0,0,3,10,0,0,0,0,0,10,3,3,0,10,3,0,0,0,0,0,3,10,0,10,0,0,0,0,0,3,0,10,3,10,3,0,0,0,0,10,0,10,0,0,3,0,0,0,0,10,3,10,3,3,0],[5,12,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,0,0,9,5,0,0,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,0,0,10,0,0,3,10,0,0,0,0,0,10,3,3,0,10,3,0,0,0,0,0,3,10,0,10,0,0,0,0,0,3,0,10,3,10,3,0,0,0,0,10,0,10,0,0,3,0,0,0,0,10,3,10,3,3,0] >;

C2×He36D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_6D_4
% in TeX

G:=Group("C2xHe3:6D4");
// GroupNames label

G:=SmallGroup(432,377);
// by ID

G=gap.SmallGroup(432,377);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,4037,1034,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=f*b*f=b^-1,c*d=d*c,e*c*e^-1=f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
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