Copied to
clipboard

G = C42×C3⋊S3order 288 = 25·32

Direct product of C42 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C42×C3⋊S3, C12215C2, C62.215C23, C1210(C4×S3), (C4×C12)⋊10S3, C32(S3×C42), C326(C2×C42), (C2×C12).423D6, (C6×C12).354C22, C6.62(S3×C2×C4), (C3×C12)⋊20(C2×C4), C3⋊Dic318(C2×C4), (C4×C3⋊Dic3)⋊28C2, (C3×C6).93(C22×C4), C22.9(C22×C3⋊S3), (C2×C6).232(C22×S3), (C22×C3⋊S3).111C22, (C2×C3⋊Dic3).184C22, C2.1(C2×C4×C3⋊S3), (C2×C4×C3⋊S3).31C2, (C2×C4).96(C2×C3⋊S3), (C2×C3⋊S3).50(C2×C4), SmallGroup(288,728)

Series: Derived Chief Lower central Upper central

C1C32 — C42×C3⋊S3
C1C3C32C3×C6C62C22×C3⋊S3C2×C4×C3⋊S3 — C42×C3⋊S3
C32 — C42×C3⋊S3
C1C42

Generators and relations for C42×C3⋊S3
 G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 324 conjugacy classes, 129 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×6], C4 [×6], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4 [×3], C2×C4 [×15], C23, C32, Dic3 [×24], C12 [×24], D6 [×24], C2×C6 [×4], C42, C42 [×3], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×48], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C2×C42, C3⋊Dic3 [×6], C3×C12 [×6], C2×C3⋊S3 [×6], C62, C4×Dic3 [×12], C4×C12 [×4], S3×C2×C4 [×12], C4×C3⋊S3 [×12], C2×C3⋊Dic3 [×3], C6×C12 [×3], C22×C3⋊S3, S3×C42 [×4], C4×C3⋊Dic3 [×3], C122, C2×C4×C3⋊S3 [×3], C42×C3⋊S3
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×4], C2×C4 [×18], C23, D6 [×12], C42 [×4], C22×C4 [×3], C3⋊S3, C4×S3 [×24], C22×S3 [×4], C2×C42, C2×C3⋊S3 [×3], S3×C2×C4 [×12], C4×C3⋊S3 [×6], C22×C3⋊S3, S3×C42 [×4], C2×C4×C3⋊S3 [×3], C42×C3⋊S3

Smallest permutation representation of C42×C3⋊S3
On 144 points
Generators in S144
(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)(109 110 111 112)(113 114 115 116)(117 118 119 120)(121 122 123 124)(125 126 127 128)(129 130 131 132)(133 134 135 136)(137 138 139 140)(141 142 143 144)
(1 9 54 77)(2 10 55 78)(3 11 56 79)(4 12 53 80)(5 75 129 104)(6 76 130 101)(7 73 131 102)(8 74 132 103)(13 71 123 28)(14 72 124 25)(15 69 121 26)(16 70 122 27)(17 133 41 108)(18 134 42 105)(19 135 43 106)(20 136 44 107)(21 68 120 46)(22 65 117 47)(23 66 118 48)(24 67 119 45)(29 139 114 109)(30 140 115 110)(31 137 116 111)(32 138 113 112)(33 85 143 126)(34 86 144 127)(35 87 141 128)(36 88 142 125)(37 93 61 50)(38 94 62 51)(39 95 63 52)(40 96 64 49)(57 99 89 84)(58 100 90 81)(59 97 91 82)(60 98 92 83)
(1 42 15)(2 43 16)(3 44 13)(4 41 14)(5 120 86)(6 117 87)(7 118 88)(8 119 85)(9 105 69)(10 106 70)(11 107 71)(12 108 72)(17 124 53)(18 121 54)(19 122 55)(20 123 56)(21 127 129)(22 128 130)(23 125 131)(24 126 132)(25 80 133)(26 77 134)(27 78 135)(28 79 136)(29 96 60)(30 93 57)(31 94 58)(32 95 59)(33 103 67)(34 104 68)(35 101 65)(36 102 66)(37 84 110)(38 81 111)(39 82 112)(40 83 109)(45 143 74)(46 144 75)(47 141 76)(48 142 73)(49 92 114)(50 89 115)(51 90 116)(52 91 113)(61 99 140)(62 100 137)(63 97 138)(64 98 139)
(1 117 51)(2 118 52)(3 119 49)(4 120 50)(5 115 14)(6 116 15)(7 113 16)(8 114 13)(9 47 38)(10 48 39)(11 45 40)(12 46 37)(17 127 57)(18 128 58)(19 125 59)(20 126 60)(21 93 53)(22 94 54)(23 95 55)(24 96 56)(25 104 140)(26 101 137)(27 102 138)(28 103 139)(29 123 132)(30 124 129)(31 121 130)(32 122 131)(33 98 136)(34 99 133)(35 100 134)(36 97 135)(41 86 89)(42 87 90)(43 88 91)(44 85 92)(61 80 68)(62 77 65)(63 78 66)(64 79 67)(69 76 111)(70 73 112)(71 74 109)(72 75 110)(81 105 141)(82 106 142)(83 107 143)(84 108 144)
(1 54)(2 55)(3 56)(4 53)(5 57)(6 58)(7 59)(8 60)(9 77)(10 78)(11 79)(12 80)(13 20)(14 17)(15 18)(16 19)(21 50)(22 51)(23 52)(24 49)(25 108)(26 105)(27 106)(28 107)(29 85)(30 86)(31 87)(32 88)(33 109)(34 110)(35 111)(36 112)(37 68)(38 65)(39 66)(40 67)(41 124)(42 121)(43 122)(44 123)(45 64)(46 61)(47 62)(48 63)(69 134)(70 135)(71 136)(72 133)(73 97)(74 98)(75 99)(76 100)(81 101)(82 102)(83 103)(84 104)(89 129)(90 130)(91 131)(92 132)(93 120)(94 117)(95 118)(96 119)(113 125)(114 126)(115 127)(116 128)(137 141)(138 142)(139 143)(140 144)

G:=sub<Sym(144)| (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)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,9,54,77)(2,10,55,78)(3,11,56,79)(4,12,53,80)(5,75,129,104)(6,76,130,101)(7,73,131,102)(8,74,132,103)(13,71,123,28)(14,72,124,25)(15,69,121,26)(16,70,122,27)(17,133,41,108)(18,134,42,105)(19,135,43,106)(20,136,44,107)(21,68,120,46)(22,65,117,47)(23,66,118,48)(24,67,119,45)(29,139,114,109)(30,140,115,110)(31,137,116,111)(32,138,113,112)(33,85,143,126)(34,86,144,127)(35,87,141,128)(36,88,142,125)(37,93,61,50)(38,94,62,51)(39,95,63,52)(40,96,64,49)(57,99,89,84)(58,100,90,81)(59,97,91,82)(60,98,92,83), (1,42,15)(2,43,16)(3,44,13)(4,41,14)(5,120,86)(6,117,87)(7,118,88)(8,119,85)(9,105,69)(10,106,70)(11,107,71)(12,108,72)(17,124,53)(18,121,54)(19,122,55)(20,123,56)(21,127,129)(22,128,130)(23,125,131)(24,126,132)(25,80,133)(26,77,134)(27,78,135)(28,79,136)(29,96,60)(30,93,57)(31,94,58)(32,95,59)(33,103,67)(34,104,68)(35,101,65)(36,102,66)(37,84,110)(38,81,111)(39,82,112)(40,83,109)(45,143,74)(46,144,75)(47,141,76)(48,142,73)(49,92,114)(50,89,115)(51,90,116)(52,91,113)(61,99,140)(62,100,137)(63,97,138)(64,98,139), (1,117,51)(2,118,52)(3,119,49)(4,120,50)(5,115,14)(6,116,15)(7,113,16)(8,114,13)(9,47,38)(10,48,39)(11,45,40)(12,46,37)(17,127,57)(18,128,58)(19,125,59)(20,126,60)(21,93,53)(22,94,54)(23,95,55)(24,96,56)(25,104,140)(26,101,137)(27,102,138)(28,103,139)(29,123,132)(30,124,129)(31,121,130)(32,122,131)(33,98,136)(34,99,133)(35,100,134)(36,97,135)(41,86,89)(42,87,90)(43,88,91)(44,85,92)(61,80,68)(62,77,65)(63,78,66)(64,79,67)(69,76,111)(70,73,112)(71,74,109)(72,75,110)(81,105,141)(82,106,142)(83,107,143)(84,108,144), (1,54)(2,55)(3,56)(4,53)(5,57)(6,58)(7,59)(8,60)(9,77)(10,78)(11,79)(12,80)(13,20)(14,17)(15,18)(16,19)(21,50)(22,51)(23,52)(24,49)(25,108)(26,105)(27,106)(28,107)(29,85)(30,86)(31,87)(32,88)(33,109)(34,110)(35,111)(36,112)(37,68)(38,65)(39,66)(40,67)(41,124)(42,121)(43,122)(44,123)(45,64)(46,61)(47,62)(48,63)(69,134)(70,135)(71,136)(72,133)(73,97)(74,98)(75,99)(76,100)(81,101)(82,102)(83,103)(84,104)(89,129)(90,130)(91,131)(92,132)(93,120)(94,117)(95,118)(96,119)(113,125)(114,126)(115,127)(116,128)(137,141)(138,142)(139,143)(140,144)>;

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)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144), (1,9,54,77)(2,10,55,78)(3,11,56,79)(4,12,53,80)(5,75,129,104)(6,76,130,101)(7,73,131,102)(8,74,132,103)(13,71,123,28)(14,72,124,25)(15,69,121,26)(16,70,122,27)(17,133,41,108)(18,134,42,105)(19,135,43,106)(20,136,44,107)(21,68,120,46)(22,65,117,47)(23,66,118,48)(24,67,119,45)(29,139,114,109)(30,140,115,110)(31,137,116,111)(32,138,113,112)(33,85,143,126)(34,86,144,127)(35,87,141,128)(36,88,142,125)(37,93,61,50)(38,94,62,51)(39,95,63,52)(40,96,64,49)(57,99,89,84)(58,100,90,81)(59,97,91,82)(60,98,92,83), (1,42,15)(2,43,16)(3,44,13)(4,41,14)(5,120,86)(6,117,87)(7,118,88)(8,119,85)(9,105,69)(10,106,70)(11,107,71)(12,108,72)(17,124,53)(18,121,54)(19,122,55)(20,123,56)(21,127,129)(22,128,130)(23,125,131)(24,126,132)(25,80,133)(26,77,134)(27,78,135)(28,79,136)(29,96,60)(30,93,57)(31,94,58)(32,95,59)(33,103,67)(34,104,68)(35,101,65)(36,102,66)(37,84,110)(38,81,111)(39,82,112)(40,83,109)(45,143,74)(46,144,75)(47,141,76)(48,142,73)(49,92,114)(50,89,115)(51,90,116)(52,91,113)(61,99,140)(62,100,137)(63,97,138)(64,98,139), (1,117,51)(2,118,52)(3,119,49)(4,120,50)(5,115,14)(6,116,15)(7,113,16)(8,114,13)(9,47,38)(10,48,39)(11,45,40)(12,46,37)(17,127,57)(18,128,58)(19,125,59)(20,126,60)(21,93,53)(22,94,54)(23,95,55)(24,96,56)(25,104,140)(26,101,137)(27,102,138)(28,103,139)(29,123,132)(30,124,129)(31,121,130)(32,122,131)(33,98,136)(34,99,133)(35,100,134)(36,97,135)(41,86,89)(42,87,90)(43,88,91)(44,85,92)(61,80,68)(62,77,65)(63,78,66)(64,79,67)(69,76,111)(70,73,112)(71,74,109)(72,75,110)(81,105,141)(82,106,142)(83,107,143)(84,108,144), (1,54)(2,55)(3,56)(4,53)(5,57)(6,58)(7,59)(8,60)(9,77)(10,78)(11,79)(12,80)(13,20)(14,17)(15,18)(16,19)(21,50)(22,51)(23,52)(24,49)(25,108)(26,105)(27,106)(28,107)(29,85)(30,86)(31,87)(32,88)(33,109)(34,110)(35,111)(36,112)(37,68)(38,65)(39,66)(40,67)(41,124)(42,121)(43,122)(44,123)(45,64)(46,61)(47,62)(48,63)(69,134)(70,135)(71,136)(72,133)(73,97)(74,98)(75,99)(76,100)(81,101)(82,102)(83,103)(84,104)(89,129)(90,130)(91,131)(92,132)(93,120)(94,117)(95,118)(96,119)(113,125)(114,126)(115,127)(116,128)(137,141)(138,142)(139,143)(140,144) );

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),(109,110,111,112),(113,114,115,116),(117,118,119,120),(121,122,123,124),(125,126,127,128),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144)], [(1,9,54,77),(2,10,55,78),(3,11,56,79),(4,12,53,80),(5,75,129,104),(6,76,130,101),(7,73,131,102),(8,74,132,103),(13,71,123,28),(14,72,124,25),(15,69,121,26),(16,70,122,27),(17,133,41,108),(18,134,42,105),(19,135,43,106),(20,136,44,107),(21,68,120,46),(22,65,117,47),(23,66,118,48),(24,67,119,45),(29,139,114,109),(30,140,115,110),(31,137,116,111),(32,138,113,112),(33,85,143,126),(34,86,144,127),(35,87,141,128),(36,88,142,125),(37,93,61,50),(38,94,62,51),(39,95,63,52),(40,96,64,49),(57,99,89,84),(58,100,90,81),(59,97,91,82),(60,98,92,83)], [(1,42,15),(2,43,16),(3,44,13),(4,41,14),(5,120,86),(6,117,87),(7,118,88),(8,119,85),(9,105,69),(10,106,70),(11,107,71),(12,108,72),(17,124,53),(18,121,54),(19,122,55),(20,123,56),(21,127,129),(22,128,130),(23,125,131),(24,126,132),(25,80,133),(26,77,134),(27,78,135),(28,79,136),(29,96,60),(30,93,57),(31,94,58),(32,95,59),(33,103,67),(34,104,68),(35,101,65),(36,102,66),(37,84,110),(38,81,111),(39,82,112),(40,83,109),(45,143,74),(46,144,75),(47,141,76),(48,142,73),(49,92,114),(50,89,115),(51,90,116),(52,91,113),(61,99,140),(62,100,137),(63,97,138),(64,98,139)], [(1,117,51),(2,118,52),(3,119,49),(4,120,50),(5,115,14),(6,116,15),(7,113,16),(8,114,13),(9,47,38),(10,48,39),(11,45,40),(12,46,37),(17,127,57),(18,128,58),(19,125,59),(20,126,60),(21,93,53),(22,94,54),(23,95,55),(24,96,56),(25,104,140),(26,101,137),(27,102,138),(28,103,139),(29,123,132),(30,124,129),(31,121,130),(32,122,131),(33,98,136),(34,99,133),(35,100,134),(36,97,135),(41,86,89),(42,87,90),(43,88,91),(44,85,92),(61,80,68),(62,77,65),(63,78,66),(64,79,67),(69,76,111),(70,73,112),(71,74,109),(72,75,110),(81,105,141),(82,106,142),(83,107,143),(84,108,144)], [(1,54),(2,55),(3,56),(4,53),(5,57),(6,58),(7,59),(8,60),(9,77),(10,78),(11,79),(12,80),(13,20),(14,17),(15,18),(16,19),(21,50),(22,51),(23,52),(24,49),(25,108),(26,105),(27,106),(28,107),(29,85),(30,86),(31,87),(32,88),(33,109),(34,110),(35,111),(36,112),(37,68),(38,65),(39,66),(40,67),(41,124),(42,121),(43,122),(44,123),(45,64),(46,61),(47,62),(48,63),(69,134),(70,135),(71,136),(72,133),(73,97),(74,98),(75,99),(76,100),(81,101),(82,102),(83,103),(84,104),(89,129),(90,130),(91,131),(92,132),(93,120),(94,117),(95,118),(96,119),(113,125),(114,126),(115,127),(116,128),(137,141),(138,142),(139,143),(140,144)])

96 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A···4L4M···4X6A···6L12A···12AV
order1222222233334···44···46···612···12
size1111999922221···19···92···22···2

96 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4S3D6C4×S3
kernelC42×C3⋊S3C4×C3⋊Dic3C122C2×C4×C3⋊S3C4×C3⋊S3C4×C12C2×C12C12
# reps13132441248

Matrix representation of C42×C3⋊S3 in GL6(𝔽13)

1200000
080000
001000
000100
000010
000001
,
500000
0120000
0012000
0001200
000010
000001
,
100000
010000
000100
00121200
000001
00001212
,
100000
010000
00121200
001000
000010
000001
,
100000
0120000
001000
00121200
000011
0000012

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12] >;

C42×C3⋊S3 in GAP, Magma, Sage, TeX

C_4^2\times C_3\rtimes S_3
% in TeX

G:=Group("C4^2xC3:S3");
// GroupNames label

G:=SmallGroup(288,728);
// by ID

G=gap.SmallGroup(288,728);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,58,2693,9414]);
// Polycyclic

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

׿
×
𝔽