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## G = C32×C22⋊A4order 432 = 24·33

### Direct product of C32 and C22⋊A4

Aliases: C32×C22⋊A4, C623A4, C243C33, C222(C32×A4), (C22×C62)⋊5C3, (C23×C6)⋊3C32, (C2×C6)⋊2(C3×A4), SmallGroup(432,771)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C32×C22⋊A4
 Chief series C1 — C22 — C24 — C22⋊A4 — C3×C22⋊A4 — C32×C22⋊A4
 Lower central C24 — C32×C22⋊A4
 Upper central C1 — C32

Generators and relations for C32×C22⋊A4
G = < a,b,c,d,e,f,g | a3=b3=c2=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >

Subgroups: 1216 in 316 conjugacy classes, 64 normal (5 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C32, C32, A4, C2×C6, C2×C6, C24, C3×C6, C22×C6, C33, C3×A4, C62, C62, C22⋊A4, C23×C6, C2×C62, C32×A4, C3×C22⋊A4, C22×C62, C32×C22⋊A4
Quotients: C1, C3, C32, A4, C33, C3×A4, C22⋊A4, C32×A4, C3×C22⋊A4, C32×C22⋊A4

Smallest permutation representation of C32×C22⋊A4
On 108 points
Generators in S108
(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 38 32)(2 39 33)(3 37 31)(4 36 21)(5 34 19)(6 35 20)(7 29 23)(8 30 24)(9 28 22)(10 16 49)(11 17 50)(12 18 51)(13 105 78)(14 103 76)(15 104 77)(25 84 90)(26 82 88)(27 83 89)(40 73 67)(41 74 68)(42 75 69)(43 65 59)(44 66 60)(45 64 58)(46 52 85)(47 53 86)(48 54 87)(55 63 70)(56 61 71)(57 62 72)(79 101 95)(80 102 96)(81 100 94)(91 99 106)(92 97 107)(93 98 108)
(1 20)(2 21)(3 19)(4 39)(5 37)(6 38)(7 50)(8 51)(9 49)(10 28)(11 29)(12 30)(13 25)(14 26)(15 27)(16 22)(17 23)(18 24)(31 34)(32 35)(33 36)(40 58)(41 59)(42 60)(43 74)(44 75)(45 73)(46 70)(47 71)(48 72)(52 55)(53 56)(54 57)(61 86)(62 87)(63 85)(64 67)(65 68)(66 69)(76 88)(77 89)(78 90)(79 97)(80 98)(81 99)(82 103)(83 104)(84 105)(91 94)(92 95)(93 96)(100 106)(101 107)(102 108)
(1 23)(2 24)(3 22)(4 51)(5 49)(6 50)(7 38)(8 39)(9 37)(10 34)(11 35)(12 36)(13 98)(14 99)(15 97)(16 19)(17 20)(18 21)(25 80)(26 81)(27 79)(28 31)(29 32)(30 33)(40 52)(41 53)(42 54)(43 61)(44 62)(45 63)(46 67)(47 68)(48 69)(55 58)(56 59)(57 60)(64 70)(65 71)(66 72)(73 85)(74 86)(75 87)(76 91)(77 92)(78 93)(82 100)(83 101)(84 102)(88 94)(89 95)(90 96)(103 106)(104 107)(105 108)
(1 20)(2 21)(3 19)(4 39)(5 37)(6 38)(7 50)(8 51)(9 49)(10 28)(11 29)(12 30)(16 22)(17 23)(18 24)(31 34)(32 35)(33 36)(40 55)(41 56)(42 57)(43 86)(44 87)(45 85)(46 64)(47 65)(48 66)(52 58)(53 59)(54 60)(61 74)(62 75)(63 73)(67 70)(68 71)(69 72)
(1 20)(2 21)(3 19)(4 39)(5 37)(6 38)(7 50)(8 51)(9 49)(10 28)(11 29)(12 30)(13 98)(14 99)(15 97)(16 22)(17 23)(18 24)(25 80)(26 81)(27 79)(31 34)(32 35)(33 36)(76 91)(77 92)(78 93)(82 100)(83 101)(84 102)(88 94)(89 95)(90 96)(103 106)(104 107)(105 108)
(1 105 73)(2 103 74)(3 104 75)(4 91 71)(5 92 72)(6 93 70)(7 96 64)(8 94 65)(9 95 66)(10 27 54)(11 25 52)(12 26 53)(13 40 32)(14 41 33)(15 42 31)(16 83 87)(17 84 85)(18 82 86)(19 107 62)(20 108 63)(21 106 61)(22 101 44)(23 102 45)(24 100 43)(28 79 60)(29 80 58)(30 81 59)(34 97 57)(35 98 55)(36 99 56)(37 77 69)(38 78 67)(39 76 68)(46 50 90)(47 51 88)(48 49 89)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2E 3A ··· 3H 3I ··· 3Z 6A ··· 6AN order 1 2 ··· 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 ··· 3 1 ··· 1 16 ··· 16 3 ··· 3

72 irreducible representations

 dim 1 1 1 3 3 type + + image C1 C3 C3 A4 C3×A4 kernel C32×C22⋊A4 C3×C22⋊A4 C22×C62 C62 C2×C6 # reps 1 24 2 5 40

Matrix representation of C32×C22⋊A4 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6
,
 0 4 0 0 0 0 0 0 4 0 0 0 4 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(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,6],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,6],[0,0,4,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C32×C22⋊A4 in GAP, Magma, Sage, TeX

C_3^2\times C_2^2\rtimes A_4
% in TeX

G:=Group("C3^2xC2^2:A4");
// GroupNames label

G:=SmallGroup(432,771);
// by ID

G=gap.SmallGroup(432,771);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,1515,2839,9077,15882]);
// Polycyclic

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

׿
×
𝔽