direct product, non-abelian, supersoluble, monomial
Aliases: C4×He3⋊3C4, He3⋊4C42, C62.28D6, (C4×He3)⋊6C4, (C6×C12).21S3, (C3×C12)⋊4Dic3, C32⋊3(C4×Dic3), C12.14(C3⋊Dic3), (C22×He3).21C22, C6.22(C4×C3⋊S3), (C3×C6).22(C4×S3), (C2×C4×He3).13C2, C3.2(C4×C3⋊Dic3), C6.18(C2×C3⋊Dic3), C2.2(C4×He3⋊C2), C2.2(C2×He3⋊3C4), (C2×C12).29(C3⋊S3), (C2×He3).23(C2×C4), (C3×C6).18(C2×Dic3), (C2×He3⋊3C4).10C2, (C2×C4).6(He3⋊C2), C22.3(C2×He3⋊C2), (C2×C6).51(C2×C3⋊S3), SmallGroup(432,186)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C4×He3⋊3C4 |
Generators and relations for C4×He3⋊3C4
G = < a,b,c,d,e | a4=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 453 in 165 conjugacy classes, 63 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C2×He3, C2×He3, C6×Dic3, C6×C12, He3⋊3C4, C4×He3, C22×He3, Dic3×C12, C2×He3⋊3C4, C2×C4×He3, C4×He3⋊3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4×Dic3, He3⋊C2, C4×C3⋊S3, C2×C3⋊Dic3, He3⋊3C4, C2×He3⋊C2, C4×C3⋊Dic3, C4×He3⋊C2, C2×He3⋊3C4, C4×He3⋊3C4
►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)
(5 63 66)(6 64 67)(7 61 68)(8 62 65)(9 141 130)(10 142 131)(11 143 132)(12 144 129)(17 32 50)(18 29 51)(19 30 52)(20 31 49)(41 56 83)(42 53 84)(43 54 81)(44 55 82)(45 128 80)(46 125 77)(47 126 78)(48 127 79)(57 72 109)(58 69 110)(59 70 111)(60 71 112)(89 122 104)(90 123 101)(91 124 102)(92 121 103)(117 140 133)(118 137 134)(119 138 135)(120 139 136)
(1 26 15)(2 27 16)(3 28 13)(4 25 14)(5 66 63)(6 67 64)(7 68 61)(8 65 62)(9 130 141)(10 131 142)(11 132 143)(12 129 144)(17 32 50)(18 29 51)(19 30 52)(20 31 49)(21 36 73)(22 33 74)(23 34 75)(24 35 76)(37 95 106)(38 96 107)(39 93 108)(40 94 105)(41 83 56)(42 84 53)(43 81 54)(44 82 55)(45 128 80)(46 125 77)(47 126 78)(48 127 79)(57 72 109)(58 69 110)(59 70 111)(60 71 112)(85 116 100)(86 113 97)(87 114 98)(88 115 99)(89 104 122)(90 101 123)(91 102 124)(92 103 121)(117 140 133)(118 137 134)(119 138 135)(120 139 136)
(1 62 29)(2 63 30)(3 64 31)(4 61 32)(5 52 27)(6 49 28)(7 50 25)(8 51 26)(9 128 105)(10 125 106)(11 126 107)(12 127 108)(13 67 20)(14 68 17)(15 65 18)(16 66 19)(21 82 72)(22 83 69)(23 84 70)(24 81 71)(33 56 110)(34 53 111)(35 54 112)(36 55 109)(37 131 77)(38 132 78)(39 129 79)(40 130 80)(41 58 74)(42 59 75)(43 60 76)(44 57 73)(45 94 141)(46 95 142)(47 96 143)(48 93 144)(85 92 139)(86 89 140)(87 90 137)(88 91 138)(97 122 117)(98 123 118)(99 124 119)(100 121 120)(101 134 114)(102 135 115)(103 136 116)(104 133 113)
(1 39 76 115)(2 40 73 116)(3 37 74 113)(4 38 75 114)(5 45 82 120)(6 46 83 117)(7 47 84 118)(8 48 81 119)(9 109 92 19)(10 110 89 20)(11 111 90 17)(12 112 91 18)(13 106 33 86)(14 107 34 87)(15 108 35 88)(16 105 36 85)(21 100 27 94)(22 97 28 95)(23 98 25 96)(24 99 26 93)(29 129 60 102)(30 130 57 103)(31 131 58 104)(32 132 59 101)(41 133 64 77)(42 134 61 78)(43 135 62 79)(44 136 63 80)(49 142 69 122)(50 143 70 123)(51 144 71 124)(52 141 72 121)(53 137 68 126)(54 138 65 127)(55 139 66 128)(56 140 67 125)
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), (5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,141,130)(10,142,131)(11,143,132)(12,144,129)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(41,56,83)(42,53,84)(43,54,81)(44,55,82)(45,128,80)(46,125,77)(47,126,78)(48,127,79)(57,72,109)(58,69,110)(59,70,111)(60,71,112)(89,122,104)(90,123,101)(91,124,102)(92,121,103)(117,140,133)(118,137,134)(119,138,135)(120,139,136), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,130,141)(10,131,142)(11,132,143)(12,129,144)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,73)(22,33,74)(23,34,75)(24,35,76)(37,95,106)(38,96,107)(39,93,108)(40,94,105)(41,83,56)(42,84,53)(43,81,54)(44,82,55)(45,128,80)(46,125,77)(47,126,78)(48,127,79)(57,72,109)(58,69,110)(59,70,111)(60,71,112)(85,116,100)(86,113,97)(87,114,98)(88,115,99)(89,104,122)(90,101,123)(91,102,124)(92,103,121)(117,140,133)(118,137,134)(119,138,135)(120,139,136), (1,62,29)(2,63,30)(3,64,31)(4,61,32)(5,52,27)(6,49,28)(7,50,25)(8,51,26)(9,128,105)(10,125,106)(11,126,107)(12,127,108)(13,67,20)(14,68,17)(15,65,18)(16,66,19)(21,82,72)(22,83,69)(23,84,70)(24,81,71)(33,56,110)(34,53,111)(35,54,112)(36,55,109)(37,131,77)(38,132,78)(39,129,79)(40,130,80)(41,58,74)(42,59,75)(43,60,76)(44,57,73)(45,94,141)(46,95,142)(47,96,143)(48,93,144)(85,92,139)(86,89,140)(87,90,137)(88,91,138)(97,122,117)(98,123,118)(99,124,119)(100,121,120)(101,134,114)(102,135,115)(103,136,116)(104,133,113), (1,39,76,115)(2,40,73,116)(3,37,74,113)(4,38,75,114)(5,45,82,120)(6,46,83,117)(7,47,84,118)(8,48,81,119)(9,109,92,19)(10,110,89,20)(11,111,90,17)(12,112,91,18)(13,106,33,86)(14,107,34,87)(15,108,35,88)(16,105,36,85)(21,100,27,94)(22,97,28,95)(23,98,25,96)(24,99,26,93)(29,129,60,102)(30,130,57,103)(31,131,58,104)(32,132,59,101)(41,133,64,77)(42,134,61,78)(43,135,62,79)(44,136,63,80)(49,142,69,122)(50,143,70,123)(51,144,71,124)(52,141,72,121)(53,137,68,126)(54,138,65,127)(55,139,66,128)(56,140,67,125)>;
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), (5,63,66)(6,64,67)(7,61,68)(8,62,65)(9,141,130)(10,142,131)(11,143,132)(12,144,129)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(41,56,83)(42,53,84)(43,54,81)(44,55,82)(45,128,80)(46,125,77)(47,126,78)(48,127,79)(57,72,109)(58,69,110)(59,70,111)(60,71,112)(89,122,104)(90,123,101)(91,124,102)(92,121,103)(117,140,133)(118,137,134)(119,138,135)(120,139,136), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,130,141)(10,131,142)(11,132,143)(12,129,144)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,73)(22,33,74)(23,34,75)(24,35,76)(37,95,106)(38,96,107)(39,93,108)(40,94,105)(41,83,56)(42,84,53)(43,81,54)(44,82,55)(45,128,80)(46,125,77)(47,126,78)(48,127,79)(57,72,109)(58,69,110)(59,70,111)(60,71,112)(85,116,100)(86,113,97)(87,114,98)(88,115,99)(89,104,122)(90,101,123)(91,102,124)(92,103,121)(117,140,133)(118,137,134)(119,138,135)(120,139,136), (1,62,29)(2,63,30)(3,64,31)(4,61,32)(5,52,27)(6,49,28)(7,50,25)(8,51,26)(9,128,105)(10,125,106)(11,126,107)(12,127,108)(13,67,20)(14,68,17)(15,65,18)(16,66,19)(21,82,72)(22,83,69)(23,84,70)(24,81,71)(33,56,110)(34,53,111)(35,54,112)(36,55,109)(37,131,77)(38,132,78)(39,129,79)(40,130,80)(41,58,74)(42,59,75)(43,60,76)(44,57,73)(45,94,141)(46,95,142)(47,96,143)(48,93,144)(85,92,139)(86,89,140)(87,90,137)(88,91,138)(97,122,117)(98,123,118)(99,124,119)(100,121,120)(101,134,114)(102,135,115)(103,136,116)(104,133,113), (1,39,76,115)(2,40,73,116)(3,37,74,113)(4,38,75,114)(5,45,82,120)(6,46,83,117)(7,47,84,118)(8,48,81,119)(9,109,92,19)(10,110,89,20)(11,111,90,17)(12,112,91,18)(13,106,33,86)(14,107,34,87)(15,108,35,88)(16,105,36,85)(21,100,27,94)(22,97,28,95)(23,98,25,96)(24,99,26,93)(29,129,60,102)(30,130,57,103)(31,131,58,104)(32,132,59,101)(41,133,64,77)(42,134,61,78)(43,135,62,79)(44,136,63,80)(49,142,69,122)(50,143,70,123)(51,144,71,124)(52,141,72,121)(53,137,68,126)(54,138,65,127)(55,139,66,128)(56,140,67,125) );
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)], [(5,63,66),(6,64,67),(7,61,68),(8,62,65),(9,141,130),(10,142,131),(11,143,132),(12,144,129),(17,32,50),(18,29,51),(19,30,52),(20,31,49),(41,56,83),(42,53,84),(43,54,81),(44,55,82),(45,128,80),(46,125,77),(47,126,78),(48,127,79),(57,72,109),(58,69,110),(59,70,111),(60,71,112),(89,122,104),(90,123,101),(91,124,102),(92,121,103),(117,140,133),(118,137,134),(119,138,135),(120,139,136)], [(1,26,15),(2,27,16),(3,28,13),(4,25,14),(5,66,63),(6,67,64),(7,68,61),(8,65,62),(9,130,141),(10,131,142),(11,132,143),(12,129,144),(17,32,50),(18,29,51),(19,30,52),(20,31,49),(21,36,73),(22,33,74),(23,34,75),(24,35,76),(37,95,106),(38,96,107),(39,93,108),(40,94,105),(41,83,56),(42,84,53),(43,81,54),(44,82,55),(45,128,80),(46,125,77),(47,126,78),(48,127,79),(57,72,109),(58,69,110),(59,70,111),(60,71,112),(85,116,100),(86,113,97),(87,114,98),(88,115,99),(89,104,122),(90,101,123),(91,102,124),(92,103,121),(117,140,133),(118,137,134),(119,138,135),(120,139,136)], [(1,62,29),(2,63,30),(3,64,31),(4,61,32),(5,52,27),(6,49,28),(7,50,25),(8,51,26),(9,128,105),(10,125,106),(11,126,107),(12,127,108),(13,67,20),(14,68,17),(15,65,18),(16,66,19),(21,82,72),(22,83,69),(23,84,70),(24,81,71),(33,56,110),(34,53,111),(35,54,112),(36,55,109),(37,131,77),(38,132,78),(39,129,79),(40,130,80),(41,58,74),(42,59,75),(43,60,76),(44,57,73),(45,94,141),(46,95,142),(47,96,143),(48,93,144),(85,92,139),(86,89,140),(87,90,137),(88,91,138),(97,122,117),(98,123,118),(99,124,119),(100,121,120),(101,134,114),(102,135,115),(103,136,116),(104,133,113)], [(1,39,76,115),(2,40,73,116),(3,37,74,113),(4,38,75,114),(5,45,82,120),(6,46,83,117),(7,47,84,118),(8,48,81,119),(9,109,92,19),(10,110,89,20),(11,111,90,17),(12,112,91,18),(13,106,33,86),(14,107,34,87),(15,108,35,88),(16,105,36,85),(21,100,27,94),(22,97,28,95),(23,98,25,96),(24,99,26,93),(29,129,60,102),(30,130,57,103),(31,131,58,104),(32,132,59,101),(41,133,64,77),(42,134,61,78),(43,135,62,79),(44,136,63,80),(49,142,69,122),(50,143,70,123),(51,144,71,124),(52,141,72,121),(53,137,68,126),(54,138,65,127),(55,139,66,128),(56,140,67,125)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6R | 12A | ··· | 12H | 12I | ··· | 12X | 12Y | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | C4×S3 | He3⋊C2 | He3⋊3C4 | C2×He3⋊C2 | C4×He3⋊C2 |
| kernel | C4×He3⋊3C4 | C2×He3⋊3C4 | C2×C4×He3 | He3⋊3C4 | C4×He3 | C6×C12 | C3×C12 | C62 | C3×C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 16 | 4 | 8 | 4 | 16 |
Matrix representation of C4×He3⋊3C4 ►in GL5(𝔽13)
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 12 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[0,1,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,12,12,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0] >;
C4×He3⋊3C4 in GAP, Magma, Sage, TeX
C_4\times {\rm He}_3\rtimes_3C_4 % in TeX
G:=Group("C4xHe3:3C4"); // GroupNames label
G:=SmallGroup(432,186);
// by ID
G=gap.SmallGroup(432,186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,64,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations