Copied to
clipboard

## G = C2×Dic3×A4order 288 = 25·32

### Direct product of C2, Dic3 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×Dic3×A4
 Chief series C1 — C3 — C2×C6 — C22×C6 — C6×A4 — Dic3×A4 — C2×Dic3×A4
 Lower central C2×C6 — C2×Dic3×A4
 Upper central C1 — C22

Generators and relations for C2×Dic3×A4
G = < a,b,c,d,e,f | a2=b6=d2=e2=f3=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 498 in 142 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C3 [×2], C4 [×4], C22 [×2], C22 [×11], C6, C6 [×2], C6 [×10], C2×C4 [×10], C23, C23 [×2], C23 [×4], C32, Dic3 [×2], Dic3 [×2], C12 [×2], A4, A4, C2×C6 [×2], C2×C6 [×13], C22×C4 [×6], C24, C3×C6 [×3], C2×Dic3, C2×Dic3 [×9], C2×C12, C2×A4, C2×A4 [×2], C2×A4 [×3], C22×C6, C22×C6 [×2], C22×C6 [×4], C23×C4, C3×Dic3 [×2], C3×A4, C62, C4×A4 [×2], C22×Dic3 [×2], C22×Dic3 [×4], C22×A4, C22×A4, C23×C6, C6×Dic3, C6×A4, C6×A4 [×2], C2×C4×A4, C23×Dic3, Dic3×A4 [×2], A4×C2×C6, C2×Dic3×A4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], A4, D6, C2×C6, C3×S3, C2×Dic3, C2×C12, C2×A4 [×3], C3×Dic3 [×2], S3×C6, C4×A4 [×2], C22×A4, C6×Dic3, S3×A4, C2×C4×A4, Dic3×A4 [×2], C2×S3×A4, C2×Dic3×A4

Smallest permutation representation of C2×Dic3×A4
On 72 points
Generators in S72
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 69)(8 70)(9 71)(10 72)(11 67)(12 68)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(55 66)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 45 4 48)(2 44 5 47)(3 43 6 46)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 53 16 50)(14 52 17 49)(15 51 18 54)(19 56 22 59)(20 55 23 58)(21 60 24 57)(25 65 28 62)(26 64 29 61)(27 63 30 66)(31 68 34 71)(32 67 35 70)(33 72 36 69)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(55 66)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 69)(8 70)(9 71)(10 72)(11 67)(12 68)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 36 24)(2 31 19)(3 32 20)(4 33 21)(5 34 22)(6 35 23)(7 62 50)(8 63 51)(9 64 52)(10 65 53)(11 66 54)(12 61 49)(13 42 25)(14 37 26)(15 38 27)(16 39 28)(17 40 29)(18 41 30)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M 6N ··· 6S 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 1 1 3 3 3 3 2 4 4 8 8 3 3 3 3 9 9 9 9 2 2 2 4 ··· 4 6 6 6 6 8 ··· 8 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 6 6 type + + + + - + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 A4 C2×A4 C2×A4 C4×A4 S3×A4 Dic3×A4 C2×S3×A4 kernel C2×Dic3×A4 Dic3×A4 A4×C2×C6 C23×Dic3 C6×A4 C22×Dic3 C23×C6 C22×C6 C22×A4 C2×A4 C2×A4 C24 C23 C23 C2×Dic3 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 2 4 2 1 2 1 4 1 2 1

Matrix representation of C2×Dic3×A4 in GL5(𝔽13)

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

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

C2×Dic3×A4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\times A_4
% in TeX

G:=Group("C2xDic3xA4");
// GroupNames label

G:=SmallGroup(288,927);
// by ID

G=gap.SmallGroup(288,927);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,84,648,271,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=d^2=e^2=f^3=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

׿
×
𝔽