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## G = C2×C4×C3.A4order 288 = 25·32

### Direct product of C2×C4 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C4×C3.A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C2×C3.A4 — C22×C3.A4 — C2×C4×C3.A4
 Lower central C22 — C2×C4×C3.A4
 Upper central C1 — C2×C12

Generators and relations for C2×C4×C3.A4
G = < a,b,c,d,e,f | a2=b4=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 276 in 116 conjugacy classes, 40 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×11], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×9], C23, C23 [×2], C23 [×4], C9, C12 [×2], C12 [×2], C2×C6 [×2], C2×C6 [×11], C22×C4 [×2], C22×C4 [×4], C24, C18 [×3], C2×C12, C2×C12 [×9], C22×C6, C22×C6 [×2], C22×C6 [×4], C23×C4, C36 [×2], C3.A4, C2×C18, C22×C12 [×2], C22×C12 [×4], C23×C6, C2×C36, C2×C3.A4, C2×C3.A4 [×2], C23×C12, C4×C3.A4 [×2], C22×C3.A4, C2×C4×C3.A4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C9, C12 [×2], A4, C2×C6, C18 [×3], C2×C12, C2×A4 [×3], C36 [×2], C3.A4, C2×C18, C4×A4 [×2], C22×A4, C2×C36, C2×C3.A4 [×3], C2×C4×A4, C4×C3.A4 [×2], C22×C3.A4, C2×C4×C3.A4

Smallest permutation representation of C2×C4×C3.A4
On 72 points
Generators in S72
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 19)(18 20)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 64)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 46)(45 47)
(1 69 50 23)(2 70 51 24)(3 71 52 25)(4 72 53 26)(5 64 54 27)(6 65 46 19)(7 66 47 20)(8 67 48 21)(9 68 49 22)(10 57 30 37)(11 58 31 38)(12 59 32 39)(13 60 33 40)(14 61 34 41)(15 62 35 42)(16 63 36 43)(17 55 28 44)(18 56 29 45)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 50)(2 60)(3 41)(4 53)(5 63)(6 44)(7 47)(8 57)(9 38)(10 21)(11 68)(12 32)(13 24)(14 71)(15 35)(16 27)(17 65)(18 29)(19 28)(20 66)(22 31)(23 69)(25 34)(26 72)(30 67)(33 70)(36 64)(37 48)(39 59)(40 51)(42 62)(43 54)(45 56)(46 55)(49 58)(52 61)
(1 39)(2 51)(3 61)(4 42)(5 54)(6 55)(7 45)(8 48)(9 58)(10 30)(11 22)(12 69)(13 33)(14 25)(15 72)(16 36)(17 19)(18 66)(20 29)(21 67)(23 32)(24 70)(26 35)(27 64)(28 65)(31 68)(34 71)(37 57)(38 49)(40 60)(41 52)(43 63)(44 46)(47 56)(50 59)(53 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)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6N 9A ··· 9F 12A ··· 12H 12I ··· 12P 18A ··· 18R 36A ··· 36X order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 3 3 3 3 1 1 1 1 1 1 3 3 3 3 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 A4 C2×A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C2×C3.A4 C4×C3.A4 kernel C2×C4×C3.A4 C4×C3.A4 C22×C3.A4 C23×C12 C2×C3.A4 C22×C12 C23×C6 C23×C4 C22×C6 C22×C4 C24 C23 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 24 1 2 1 2 4 4 2 8

Matrix representation of C2×C4×C3.A4 in GL4(𝔽37) generated by

 36 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 6 0 0 0 0 31 0 0 0 0 31 0 0 0 0 31
,
 26 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 36 0 4 0 0 36 0 0 0 0 1
,
 1 0 0 0 0 36 25 0 0 0 1 0 0 0 0 36
,
 9 0 0 0 0 34 0 0 0 0 0 1 0 17 28 3
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[26,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,36,0,0,0,0,36,0,0,4,0,1],[1,0,0,0,0,36,0,0,0,25,1,0,0,0,0,36],[9,0,0,0,0,34,0,17,0,0,0,28,0,0,1,3] >;

C2×C4×C3.A4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3.A_4
% in TeX

G:=Group("C2xC4xC3.A4");
// GroupNames label

G:=SmallGroup(288,343);
// by ID

G=gap.SmallGroup(288,343);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,92,142,1531,2666]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=c^3=d^2=e^2=1,f^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,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

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