Copied to
clipboard

## G = He3⋊7Q16order 432 = 24·33

### 2nd semidirect product of He3 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×He3 — He3⋊7Q16
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4Q8 — He3⋊7Q16
 Lower central He3 — C2×He3 — C4×He3 — He3⋊7Q16
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for He37Q16
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, He33C4, C4×He3, C4×He3, C3×C3⋊Q16, He34C8, He34Q8, Q8×He3, He37Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊S3, C3⋊D4, C2×C3⋊S3, C3⋊Q16, He3⋊C2, C327D4, C2×He3⋊C2, C327Q16, He37D4, He37Q16

Smallest permutation representation of He37Q16
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 He37Q16 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] >;

He37Q16 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

׿
×
𝔽