Copied to
clipboard

## G = C22×C9⋊A4order 432 = 24·33

### Direct product of C22 and C9⋊A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C22×C9⋊A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9⋊A4 — C2×C9⋊A4 — C22×C9⋊A4
 Lower central C22 — C2×C6 — C22×C9⋊A4
 Upper central C1 — C2×C6 — C2×C18

Generators and relations for C22×C9⋊A4
G = < a,b,c,d,e,f | a2=b2=c9=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=c7, fdf-1=de=ed, fef-1=d >

Subgroups: 376 in 137 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C9, C9, C32, A4, C2×C6, C2×C6, C24, C18, C18, C3×C6, C2×A4, C22×C6, C22×C6, 3- 1+2, C3.A4, C2×C18, C2×C18, C3×A4, C62, C22×A4, C23×C6, C2×3- 1+2, C2×C3.A4, C22×C18, C22×C18, C6×A4, C9⋊A4, C22×3- 1+2, C22×C3.A4, C23×C18, A4×C2×C6, C2×C9⋊A4, C22×C9⋊A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, 3- 1+2, C3×A4, C62, C22×A4, C2×3- 1+2, C6×A4, C9⋊A4, C22×3- 1+2, A4×C2×C6, C2×C9⋊A4, C22×C9⋊A4

Smallest permutation representation of C22×C9⋊A4
On 108 points
Generators in S108
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 104)(29 105)(30 106)(31 107)(32 108)(33 100)(34 101)(35 102)(36 103)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 46)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 82)(71 83)(72 84)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 91)(81 92)
(1 54)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)(19 104)(20 105)(21 106)(22 107)(23 108)(24 100)(25 101)(26 102)(27 103)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 55)(45 56)(64 96)(65 97)(66 98)(67 99)(68 91)(69 92)(70 93)(71 94)(72 95)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)
(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 54)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(28 104)(29 105)(30 106)(31 107)(32 108)(33 100)(34 101)(35 102)(36 103)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 55)(45 56)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 73)(71 74)(72 75)(82 93)(83 94)(84 95)(85 96)(86 97)(87 98)(88 99)(89 91)(90 92)
(1 44)(2 45)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)(19 104)(20 105)(21 106)(22 107)(23 108)(24 100)(25 101)(26 102)(27 103)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 55)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 82)(71 83)(72 84)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 91)(81 92)
(1 105 69)(2 100 67)(3 104 65)(4 108 72)(5 103 70)(6 107 68)(7 102 66)(8 106 64)(9 101 71)(10 79 45)(11 74 43)(12 78 41)(13 73 39)(14 77 37)(15 81 44)(16 76 42)(17 80 40)(18 75 38)(19 97 47)(20 92 54)(21 96 52)(22 91 50)(23 95 48)(24 99 46)(25 94 53)(26 98 51)(27 93 49)(28 86 57)(29 90 55)(30 85 62)(31 89 60)(32 84 58)(33 88 56)(34 83 63)(35 87 61)(36 82 59)

G:=sub<Sym(108)| (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,104)(29,105)(30,106)(31,107)(32,108)(33,100)(34,101)(35,102)(36,103)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,46)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,91)(81,92), (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,33)(11,34)(12,35)(13,36)(14,28)(15,29)(16,30)(17,31)(18,32)(19,104)(20,105)(21,106)(22,107)(23,108)(24,100)(25,101)(26,102)(27,103)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56)(64,96)(65,97)(66,98)(67,99)(68,91)(69,92)(70,93)(71,94)(72,95)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,104)(29,105)(30,106)(31,107)(32,108)(33,100)(34,101)(35,102)(36,103)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,73)(71,74)(72,75)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,91)(90,92), (1,44)(2,45)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,33)(11,34)(12,35)(13,36)(14,28)(15,29)(16,30)(17,31)(18,32)(19,104)(20,105)(21,106)(22,107)(23,108)(24,100)(25,101)(26,102)(27,103)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,55)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,91)(81,92), (1,105,69)(2,100,67)(3,104,65)(4,108,72)(5,103,70)(6,107,68)(7,102,66)(8,106,64)(9,101,71)(10,79,45)(11,74,43)(12,78,41)(13,73,39)(14,77,37)(15,81,44)(16,76,42)(17,80,40)(18,75,38)(19,97,47)(20,92,54)(21,96,52)(22,91,50)(23,95,48)(24,99,46)(25,94,53)(26,98,51)(27,93,49)(28,86,57)(29,90,55)(30,85,62)(31,89,60)(32,84,58)(33,88,56)(34,83,63)(35,87,61)(36,82,59)>;

G:=Group( (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,104)(29,105)(30,106)(31,107)(32,108)(33,100)(34,101)(35,102)(36,103)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,46)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,91)(81,92), (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,33)(11,34)(12,35)(13,36)(14,28)(15,29)(16,30)(17,31)(18,32)(19,104)(20,105)(21,106)(22,107)(23,108)(24,100)(25,101)(26,102)(27,103)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56)(64,96)(65,97)(66,98)(67,99)(68,91)(69,92)(70,93)(71,94)(72,95)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90), (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(28,104)(29,105)(30,106)(31,107)(32,108)(33,100)(34,101)(35,102)(36,103)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,55)(45,56)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,73)(71,74)(72,75)(82,93)(83,94)(84,95)(85,96)(86,97)(87,98)(88,99)(89,91)(90,92), (1,44)(2,45)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,33)(11,34)(12,35)(13,36)(14,28)(15,29)(16,30)(17,31)(18,32)(19,104)(20,105)(21,106)(22,107)(23,108)(24,100)(25,101)(26,102)(27,103)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,55)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,82)(71,83)(72,84)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,91)(81,92), (1,105,69)(2,100,67)(3,104,65)(4,108,72)(5,103,70)(6,107,68)(7,102,66)(8,106,64)(9,101,71)(10,79,45)(11,74,43)(12,78,41)(13,73,39)(14,77,37)(15,81,44)(16,76,42)(17,80,40)(18,75,38)(19,97,47)(20,92,54)(21,96,52)(22,91,50)(23,95,48)(24,99,46)(25,94,53)(26,98,51)(27,93,49)(28,86,57)(29,90,55)(30,85,62)(31,89,60)(32,84,58)(33,88,56)(34,83,63)(35,87,61)(36,82,59) );

G=PermutationGroup([[(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(28,104),(29,105),(30,106),(31,107),(32,108),(33,100),(34,101),(35,102),(36,103),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,46),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,82),(71,83),(72,84),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,91),(81,92)], [(1,54),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,33),(11,34),(12,35),(13,36),(14,28),(15,29),(16,30),(17,31),(18,32),(19,104),(20,105),(21,106),(22,107),(23,108),(24,100),(25,101),(26,102),(27,103),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,55),(45,56),(64,96),(65,97),(66,98),(67,99),(68,91),(69,92),(70,93),(71,94),(72,95),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90)], [(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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,54),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(28,104),(29,105),(30,106),(31,107),(32,108),(33,100),(34,101),(35,102),(36,103),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,55),(45,56),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,73),(71,74),(72,75),(82,93),(83,94),(84,95),(85,96),(86,97),(87,98),(88,99),(89,91),(90,92)], [(1,44),(2,45),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,33),(11,34),(12,35),(13,36),(14,28),(15,29),(16,30),(17,31),(18,32),(19,104),(20,105),(21,106),(22,107),(23,108),(24,100),(25,101),(26,102),(27,103),(46,56),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,55),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,82),(71,83),(72,84),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,91),(81,92)], [(1,105,69),(2,100,67),(3,104,65),(4,108,72),(5,103,70),(6,107,68),(7,102,66),(8,106,64),(9,101,71),(10,79,45),(11,74,43),(12,78,41),(13,73,39),(14,77,37),(15,81,44),(16,76,42),(17,80,40),(18,75,38),(19,97,47),(20,92,54),(21,96,52),(22,91,50),(23,95,48),(24,99,46),(25,94,53),(26,98,51),(27,93,49),(28,86,57),(29,90,55),(30,85,62),(31,89,60),(32,84,58),(33,88,56),(34,83,63),(35,87,61),(36,82,59)]])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 6A ··· 6F 6G ··· 6N 6O ··· 6T 9A 9B 9C 9D 9E 9F 18A ··· 18AD 18AE ··· 18AP order 1 2 2 2 2 2 2 2 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 9 9 9 9 9 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 3 3 1 1 12 12 1 ··· 1 3 ··· 3 12 ··· 12 3 3 12 12 12 12 3 ··· 3 12 ··· 12

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C9⋊A4 C2×C9⋊A4 kernel C22×C9⋊A4 C2×C9⋊A4 C22×C3.A4 C23×C18 A4×C2×C6 C2×C3.A4 C22×C18 C6×A4 C2×C18 C18 C24 C2×C6 C23 C6 C22 C2 # reps 1 3 4 2 2 12 6 6 1 3 2 2 6 6 6 18

Matrix representation of C22×C9⋊A4 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 9 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 18
,
 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C22×C9⋊A4 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes A_4
% in TeX

G:=Group("C2^2xC9:A4");
// GroupNames label

G:=SmallGroup(432,547);
// by ID

G=gap.SmallGroup(432,547);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,353,108,2287,3989]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^9=d^2=e^2=f^3=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,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c^7,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

׿
×
𝔽