non-abelian, supersoluble, monomial
Aliases: He3⋊7Q16, (C3×C12).19D6, (Q8×He3).2C2, (C2×He3).37D4, He3⋊4C8.3C2, He3⋊4Q8.2C2, (Q8×C32).4S3, C2.7(He3⋊7D4), C32⋊5(C3⋊Q16), Q8.2(He3⋊C2), C6.43(C32⋊7D4), (C4×He3).15C22, C3.2(C32⋊7Q16), C12.47(C2×C3⋊S3), C4.4(C2×He3⋊C2), (C3×C6).38(C3⋊D4), (C3×Q8).15(C3⋊S3), SmallGroup(432,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊7Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 329 in 99 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C32, Dic3, C12, C12, Q16, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, He3, C3×Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C2×He3, C3×C3⋊C8, C3×Dic6, Q8×C32, He3⋊3C4, C4×He3, C4×He3, C3×C3⋊Q16, He3⋊4C8, He3⋊4Q8, Q8×He3, He3⋊7Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊S3, C3⋊D4, C2×C3⋊S3, C3⋊Q16, He3⋊C2, C32⋊7D4, C2×He3⋊C2, C32⋊7Q16, He3⋊7D4, He3⋊7Q16
►On 144 points
Generators in S144
(9 73 31)(10 32 74)(11 75 25)(12 26 76)(13 77 27)(14 28 78)(15 79 29)(16 30 80)(33 102 137)(34 138 103)(35 104 139)(36 140 97)(37 98 141)(38 142 99)(39 100 143)(40 144 101)(49 125 65)(50 66 126)(51 127 67)(52 68 128)(53 121 69)(54 70 122)(55 123 71)(56 72 124)(89 110 133)(90 134 111)(91 112 135)(92 136 105)(93 106 129)(94 130 107)(95 108 131)(96 132 109)
(1 113 18)(2 114 19)(3 115 20)(4 116 21)(5 117 22)(6 118 23)(7 119 24)(8 120 17)(9 73 31)(10 74 32)(11 75 25)(12 76 26)(13 77 27)(14 78 28)(15 79 29)(16 80 30)(33 137 102)(34 138 103)(35 139 104)(36 140 97)(37 141 98)(38 142 99)(39 143 100)(40 144 101)(41 63 88)(42 64 81)(43 57 82)(44 58 83)(45 59 84)(46 60 85)(47 61 86)(48 62 87)(49 125 65)(50 126 66)(51 127 67)(52 128 68)(53 121 69)(54 122 70)(55 123 71)(56 124 72)(89 110 133)(90 111 134)(91 112 135)(92 105 136)(93 106 129)(94 107 130)(95 108 131)(96 109 132)
(1 54 29)(2 30 55)(3 56 31)(4 32 49)(5 50 25)(6 26 51)(7 52 27)(8 28 53)(9 115 124)(10 125 116)(11 117 126)(12 127 118)(13 119 128)(14 121 120)(15 113 122)(16 123 114)(17 78 69)(18 70 79)(19 80 71)(20 72 73)(21 74 65)(22 66 75)(23 76 67)(24 68 77)(33 131 43)(34 44 132)(35 133 45)(36 46 134)(37 135 47)(38 48 136)(39 129 41)(40 42 130)(57 137 95)(58 96 138)(59 139 89)(60 90 140)(61 141 91)(62 92 142)(63 143 93)(64 94 144)(81 107 101)(82 102 108)(83 109 103)(84 104 110)(85 111 97)(86 98 112)(87 105 99)(88 100 106)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 110 13 106)(10 109 14 105)(11 108 15 112)(12 107 16 111)(17 48 21 44)(18 47 22 43)(19 46 23 42)(20 45 24 41)(25 95 29 91)(26 94 30 90)(27 93 31 89)(28 92 32 96)(33 70 37 66)(34 69 38 65)(35 68 39 72)(36 67 40 71)(49 138 53 142)(50 137 54 141)(51 144 55 140)(52 143 56 139)(73 133 77 129)(74 132 78 136)(75 131 79 135)(76 130 80 134)(81 114 85 118)(82 113 86 117)(83 120 87 116)(84 119 88 115)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)
G:=sub<Sym(144)| (9,73,31)(10,32,74)(11,75,25)(12,26,76)(13,77,27)(14,28,78)(15,79,29)(16,30,80)(33,102,137)(34,138,103)(35,104,139)(36,140,97)(37,98,141)(38,142,99)(39,100,143)(40,144,101)(49,125,65)(50,66,126)(51,127,67)(52,68,128)(53,121,69)(54,70,122)(55,123,71)(56,72,124)(89,110,133)(90,134,111)(91,112,135)(92,136,105)(93,106,129)(94,130,107)(95,108,131)(96,132,109), (1,113,18)(2,114,19)(3,115,20)(4,116,21)(5,117,22)(6,118,23)(7,119,24)(8,120,17)(9,73,31)(10,74,32)(11,75,25)(12,76,26)(13,77,27)(14,78,28)(15,79,29)(16,80,30)(33,137,102)(34,138,103)(35,139,104)(36,140,97)(37,141,98)(38,142,99)(39,143,100)(40,144,101)(41,63,88)(42,64,81)(43,57,82)(44,58,83)(45,59,84)(46,60,85)(47,61,86)(48,62,87)(49,125,65)(50,126,66)(51,127,67)(52,128,68)(53,121,69)(54,122,70)(55,123,71)(56,124,72)(89,110,133)(90,111,134)(91,112,135)(92,105,136)(93,106,129)(94,107,130)(95,108,131)(96,109,132), (1,54,29)(2,30,55)(3,56,31)(4,32,49)(5,50,25)(6,26,51)(7,52,27)(8,28,53)(9,115,124)(10,125,116)(11,117,126)(12,127,118)(13,119,128)(14,121,120)(15,113,122)(16,123,114)(17,78,69)(18,70,79)(19,80,71)(20,72,73)(21,74,65)(22,66,75)(23,76,67)(24,68,77)(33,131,43)(34,44,132)(35,133,45)(36,46,134)(37,135,47)(38,48,136)(39,129,41)(40,42,130)(57,137,95)(58,96,138)(59,139,89)(60,90,140)(61,141,91)(62,92,142)(63,143,93)(64,94,144)(81,107,101)(82,102,108)(83,109,103)(84,104,110)(85,111,97)(86,98,112)(87,105,99)(88,100,106), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,110,13,106)(10,109,14,105)(11,108,15,112)(12,107,16,111)(17,48,21,44)(18,47,22,43)(19,46,23,42)(20,45,24,41)(25,95,29,91)(26,94,30,90)(27,93,31,89)(28,92,32,96)(33,70,37,66)(34,69,38,65)(35,68,39,72)(36,67,40,71)(49,138,53,142)(50,137,54,141)(51,144,55,140)(52,143,56,139)(73,133,77,129)(74,132,78,136)(75,131,79,135)(76,130,80,134)(81,114,85,118)(82,113,86,117)(83,120,87,116)(84,119,88,115)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128)>;
G:=Group( (9,73,31)(10,32,74)(11,75,25)(12,26,76)(13,77,27)(14,28,78)(15,79,29)(16,30,80)(33,102,137)(34,138,103)(35,104,139)(36,140,97)(37,98,141)(38,142,99)(39,100,143)(40,144,101)(49,125,65)(50,66,126)(51,127,67)(52,68,128)(53,121,69)(54,70,122)(55,123,71)(56,72,124)(89,110,133)(90,134,111)(91,112,135)(92,136,105)(93,106,129)(94,130,107)(95,108,131)(96,132,109), (1,113,18)(2,114,19)(3,115,20)(4,116,21)(5,117,22)(6,118,23)(7,119,24)(8,120,17)(9,73,31)(10,74,32)(11,75,25)(12,76,26)(13,77,27)(14,78,28)(15,79,29)(16,80,30)(33,137,102)(34,138,103)(35,139,104)(36,140,97)(37,141,98)(38,142,99)(39,143,100)(40,144,101)(41,63,88)(42,64,81)(43,57,82)(44,58,83)(45,59,84)(46,60,85)(47,61,86)(48,62,87)(49,125,65)(50,126,66)(51,127,67)(52,128,68)(53,121,69)(54,122,70)(55,123,71)(56,124,72)(89,110,133)(90,111,134)(91,112,135)(92,105,136)(93,106,129)(94,107,130)(95,108,131)(96,109,132), (1,54,29)(2,30,55)(3,56,31)(4,32,49)(5,50,25)(6,26,51)(7,52,27)(8,28,53)(9,115,124)(10,125,116)(11,117,126)(12,127,118)(13,119,128)(14,121,120)(15,113,122)(16,123,114)(17,78,69)(18,70,79)(19,80,71)(20,72,73)(21,74,65)(22,66,75)(23,76,67)(24,68,77)(33,131,43)(34,44,132)(35,133,45)(36,46,134)(37,135,47)(38,48,136)(39,129,41)(40,42,130)(57,137,95)(58,96,138)(59,139,89)(60,90,140)(61,141,91)(62,92,142)(63,143,93)(64,94,144)(81,107,101)(82,102,108)(83,109,103)(84,104,110)(85,111,97)(86,98,112)(87,105,99)(88,100,106), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,110,13,106)(10,109,14,105)(11,108,15,112)(12,107,16,111)(17,48,21,44)(18,47,22,43)(19,46,23,42)(20,45,24,41)(25,95,29,91)(26,94,30,90)(27,93,31,89)(28,92,32,96)(33,70,37,66)(34,69,38,65)(35,68,39,72)(36,67,40,71)(49,138,53,142)(50,137,54,141)(51,144,55,140)(52,143,56,139)(73,133,77,129)(74,132,78,136)(75,131,79,135)(76,130,80,134)(81,114,85,118)(82,113,86,117)(83,120,87,116)(84,119,88,115)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128) );
G=PermutationGroup([[(9,73,31),(10,32,74),(11,75,25),(12,26,76),(13,77,27),(14,28,78),(15,79,29),(16,30,80),(33,102,137),(34,138,103),(35,104,139),(36,140,97),(37,98,141),(38,142,99),(39,100,143),(40,144,101),(49,125,65),(50,66,126),(51,127,67),(52,68,128),(53,121,69),(54,70,122),(55,123,71),(56,72,124),(89,110,133),(90,134,111),(91,112,135),(92,136,105),(93,106,129),(94,130,107),(95,108,131),(96,132,109)], [(1,113,18),(2,114,19),(3,115,20),(4,116,21),(5,117,22),(6,118,23),(7,119,24),(8,120,17),(9,73,31),(10,74,32),(11,75,25),(12,76,26),(13,77,27),(14,78,28),(15,79,29),(16,80,30),(33,137,102),(34,138,103),(35,139,104),(36,140,97),(37,141,98),(38,142,99),(39,143,100),(40,144,101),(41,63,88),(42,64,81),(43,57,82),(44,58,83),(45,59,84),(46,60,85),(47,61,86),(48,62,87),(49,125,65),(50,126,66),(51,127,67),(52,128,68),(53,121,69),(54,122,70),(55,123,71),(56,124,72),(89,110,133),(90,111,134),(91,112,135),(92,105,136),(93,106,129),(94,107,130),(95,108,131),(96,109,132)], [(1,54,29),(2,30,55),(3,56,31),(4,32,49),(5,50,25),(6,26,51),(7,52,27),(8,28,53),(9,115,124),(10,125,116),(11,117,126),(12,127,118),(13,119,128),(14,121,120),(15,113,122),(16,123,114),(17,78,69),(18,70,79),(19,80,71),(20,72,73),(21,74,65),(22,66,75),(23,76,67),(24,68,77),(33,131,43),(34,44,132),(35,133,45),(36,46,134),(37,135,47),(38,48,136),(39,129,41),(40,42,130),(57,137,95),(58,96,138),(59,139,89),(60,90,140),(61,141,91),(62,92,142),(63,143,93),(64,94,144),(81,107,101),(82,102,108),(83,109,103),(84,104,110),(85,111,97),(86,98,112),(87,105,99),(88,100,106)], [(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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,110,13,106),(10,109,14,105),(11,108,15,112),(12,107,16,111),(17,48,21,44),(18,47,22,43),(19,46,23,42),(20,45,24,41),(25,95,29,91),(26,94,30,90),(27,93,31,89),(28,92,32,96),(33,70,37,66),(34,69,38,65),(35,68,39,72),(36,67,40,71),(49,138,53,142),(50,137,54,141),(51,144,55,140),(52,143,56,139),(73,133,77,129),(74,132,78,136),(75,131,79,135),(76,130,80,134),(81,114,85,118),(82,113,86,117),(83,120,87,116),(84,119,88,115),(97,127,101,123),(98,126,102,122),(99,125,103,121),(100,124,104,128)]])
41 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | 12R | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 4 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 4 | 12 | ··· | 12 | 36 | 36 | 18 | 18 | 18 | 18 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | Q16 | C3⋊D4 | He3⋊C2 | C2×He3⋊C2 | C3⋊Q16 | He3⋊7D4 | He3⋊7Q16 |
| kernel | He3⋊7Q16 | He3⋊4C8 | He3⋊4Q8 | Q8×He3 | Q8×C32 | C2×He3 | C3×C12 | He3 | C3×C6 | Q8 | C4 | C32 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of He3⋊7Q16 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 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 |
| 0 | 39 | 0 | 0 | 0 |
| 58 | 32 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 49 | 68 | 0 | 0 | 0 |
| 57 | 24 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[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],[0,58,0,0,0,39,32,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[49,57,0,0,0,68,24,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;
He3⋊7Q16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_7Q_{16} % in TeX
G:=Group("He3:7Q16"); // GroupNames label
G:=SmallGroup(432,197);
// by ID
G=gap.SmallGroup(432,197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations