Copied to
clipboard

## G = C32×A4⋊C4order 432 = 24·33

### Direct product of C32 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C32×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — C6×A4 — A4×C3×C6 — C32×A4⋊C4
 Lower central A4 — C32×A4⋊C4
 Upper central C1 — C3×C6

Generators and relations for C32×A4⋊C4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 472 in 146 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, A4, A4, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×C12, C2×A4, C2×A4, C22×C6, C33, C3×Dic3, C3×C12, C3×A4, C3×A4, C62, C62, C3×C22⋊C4, A4⋊C4, C32×C6, C6×C12, C6×A4, C6×A4, C2×C62, C32×Dic3, C32×A4, C32×C22⋊C4, C3×A4⋊C4, A4×C3×C6, C32×A4⋊C4
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, S4, C3×Dic3, C3×C12, A4⋊C4, S3×C32, C3×S4, C32×Dic3, C3×A4⋊C4, C32×S4, C32×A4⋊C4

Smallest permutation representation of C32×A4⋊C4
On 108 points
Generators in S108
(1 107 41)(2 108 42)(3 105 43)(4 106 44)(5 80 69)(6 77 70)(7 78 71)(8 79 72)(9 99 20)(10 100 17)(11 97 18)(12 98 19)(13 38 30)(14 39 31)(15 40 32)(16 37 29)(21 46 34)(22 47 35)(23 48 36)(24 45 33)(25 58 49)(26 59 50)(27 60 51)(28 57 52)(53 93 81)(54 94 82)(55 95 83)(56 96 84)(61 86 73)(62 87 74)(63 88 75)(64 85 76)(65 101 89)(66 102 90)(67 103 91)(68 104 92)
(1 87 9)(2 88 10)(3 85 11)(4 86 12)(5 65 35)(6 66 36)(7 67 33)(8 68 34)(13 26 96)(14 27 93)(15 28 94)(16 25 95)(17 42 63)(18 43 64)(19 44 61)(20 41 62)(21 79 104)(22 80 101)(23 77 102)(24 78 103)(29 49 55)(30 50 56)(31 51 53)(32 52 54)(37 58 83)(38 59 84)(39 60 81)(40 57 82)(45 71 91)(46 72 92)(47 69 89)(48 70 90)(73 98 106)(74 99 107)(75 100 108)(76 97 105)
(2 4)(5 7)(6 8)(10 12)(14 16)(17 19)(21 23)(22 24)(25 27)(29 31)(33 35)(34 36)(37 39)(42 44)(45 47)(46 48)(49 51)(53 55)(58 60)(61 63)(65 67)(66 68)(69 71)(70 72)(73 75)(77 79)(78 80)(81 83)(86 88)(89 91)(90 92)(93 95)(98 100)(101 103)(102 104)(106 108)
(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)(37 39)(38 40)(41 43)(42 44)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(73 75)(74 76)(81 83)(82 84)(85 87)(86 88)(93 95)(94 96)(97 99)(98 100)(105 107)(106 108)
(1 7 83)(2 84 8)(3 5 81)(4 82 6)(9 33 58)(10 59 34)(11 35 60)(12 57 36)(13 92 63)(14 64 89)(15 90 61)(16 62 91)(17 26 46)(18 47 27)(19 28 48)(20 45 25)(21 100 50)(22 51 97)(23 98 52)(24 49 99)(29 74 103)(30 104 75)(31 76 101)(32 102 73)(37 87 67)(38 68 88)(39 85 65)(40 66 86)(41 71 95)(42 96 72)(43 69 93)(44 94 70)(53 105 80)(54 77 106)(55 107 78)(56 79 108)
(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)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6H 6I ··· 6X 6Y ··· 6AG 12A ··· 12AF order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 3 3 1 ··· 1 8 ··· 8 6 6 6 6 1 ··· 1 3 ··· 3 8 ··· 8 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 type + + + - + image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 S4 A4⋊C4 C3×S4 C3×A4⋊C4 kernel C32×A4⋊C4 A4×C3×C6 C3×A4⋊C4 C32×A4 C6×A4 C3×A4 C2×C62 C62 C22×C6 C2×C6 C3×C6 C32 C6 C3 # reps 1 1 8 2 8 16 1 1 8 8 2 2 16 16

Matrix representation of C32×A4⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 9 0 0 0 0 0 9 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 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 12 12 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 8 0 0 0 8 0 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,1,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[12,0,0,0,0,12,1,0,0,0,0,0,0,0,8,0,0,0,8,0,0,0,8,0,0] >;

C32×A4⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times A_4\rtimes C_4
% in TeX

G:=Group("C3^2xA4:C4");
// GroupNames label

G:=SmallGroup(432,615);
// by ID

G=gap.SmallGroup(432,615);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,2524,9077,285,5298,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=e^3=f^4=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,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽