Copied to
clipboard

## G = C2×He3⋊7D4order 432 = 24·33

### Direct product of C2 and He3⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C2×He3⋊7D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C22×He3⋊C2 — C2×He3⋊7D4
 Lower central He3 — C2×He3 — C2×He3⋊7D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C2×He37D4
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, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1121 in 297 conjugacy classes, 67 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, He3, C3×Dic3, S3×C6, C62, C62, C2×C3⋊D4, C6×D4, He3⋊C2, C2×He3, C2×He3, C2×He3, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, He33C4, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C22×He3, C22×He3, C6×C3⋊D4, C2×He33C4, He37D4, C22×He3⋊C2, C23×He3, C2×He37D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×C3⋊S3, C2×C3⋊D4, He3⋊C2, C327D4, C22×C3⋊S3, C2×He3⋊C2, C2×C327D4, He37D4, C22×He3⋊C2, C2×He37D4

Smallest permutation representation of C2×He37D4
On 72 points
Generators in S72
(1 23)(2 24)(3 21)(4 22)(5 31)(6 32)(7 29)(8 30)(9 57)(10 58)(11 59)(12 60)(13 37)(14 38)(15 39)(16 40)(17 61)(18 62)(19 63)(20 64)(25 68)(26 65)(27 66)(28 67)(33 47)(34 48)(35 45)(36 46)(41 70)(42 71)(43 72)(44 69)(49 56)(50 53)(51 54)(52 55)
(1 37 11)(2 12 38)(3 39 9)(4 10 40)(5 61 33)(6 34 62)(7 63 35)(8 36 64)(13 59 23)(14 24 60)(15 57 21)(16 22 58)(17 47 31)(18 32 48)(19 45 29)(20 30 46)(25 69 54)(26 55 70)(27 71 56)(28 53 72)(41 65 52)(42 49 66)(43 67 50)(44 51 68)
(1 27 5)(2 28 6)(3 25 7)(4 26 8)(9 54 35)(10 55 36)(11 56 33)(12 53 34)(13 42 17)(14 43 18)(15 44 19)(16 41 20)(21 68 29)(22 65 30)(23 66 31)(24 67 32)(37 71 61)(38 72 62)(39 69 63)(40 70 64)(45 57 51)(46 58 52)(47 59 49)(48 60 50)
(1 56 71)(2 72 53)(3 54 69)(4 70 55)(5 11 37)(6 38 12)(7 9 39)(8 40 10)(13 31 59)(14 60 32)(15 29 57)(16 58 30)(17 66 47)(18 48 67)(19 68 45)(20 46 65)(21 51 44)(22 41 52)(23 49 42)(24 43 50)(25 35 63)(26 64 36)(27 33 61)(28 62 34)
(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)
(2 4)(6 8)(9 39)(10 38)(11 37)(12 40)(13 59)(14 58)(15 57)(16 60)(17 47)(18 46)(19 45)(20 48)(22 24)(26 28)(30 32)(33 61)(34 64)(35 63)(36 62)(41 50)(42 49)(43 52)(44 51)(53 70)(54 69)(55 72)(56 71)(65 67)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A ··· 6F 6G 6H 6I 6J 6K ··· 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 6 6 12 12 12 12 size 1 1 1 1 2 2 18 18 1 1 6 6 6 6 18 18 1 ··· 1 2 2 2 2 6 ··· 6 18 18 18 18 18 18 18 18

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 3 3 6 type + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C3⋊D4 He3⋊C2 C2×He3⋊C2 He3⋊7D4 kernel C2×He3⋊7D4 C2×He3⋊3C4 He3⋊7D4 C22×He3⋊C2 C23×He3 C2×C62 C2×He3 C62 C3×C6 C23 C22 C2 # reps 1 1 4 1 1 4 2 12 16 4 12 4

Matrix representation of C2×He37D4 in GL5(𝔽13)

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

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

C2×He37D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_7D_4
% in TeX

G:=Group("C2xHe3:7D4");
// GroupNames label

G:=SmallGroup(432,399);
// by ID

G=gap.SmallGroup(432,399);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,1124,4037,537]);
// 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,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽