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## G = C2×C8×He3order 432 = 24·33

### Direct product of C2×C8 and He3

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C8×He3
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×He3 — C8×He3 — C2×C8×He3
 Lower central C1 — C3 — C2×C8×He3
 Upper central C1 — C2×C24 — C2×C8×He3

Generators and relations for C2×C8×He3
G = < a,b,c,d,e | a2=b8=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 209 in 121 conjugacy classes, 77 normal (21 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, C6, C6 [×2], C6 [×12], C8 [×2], C2×C4, C32 [×4], C12 [×2], C12 [×8], C2×C6, C2×C6 [×4], C2×C8, C3×C6 [×12], C24 [×2], C24 [×8], C2×C12, C2×C12 [×4], He3, C3×C12 [×8], C62 [×4], C2×C24, C2×C24 [×4], C2×He3, C2×He3 [×2], C3×C24 [×8], C6×C12 [×4], C4×He3 [×2], C22×He3, C6×C24 [×4], C8×He3 [×2], C2×C4×He3, C2×C8×He3
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, C3×C6 [×3], C24 [×8], C2×C12 [×4], He3, C3×C12 [×2], C62, C2×C24 [×4], C2×He3 [×3], C3×C24 [×2], C6×C12, C4×He3 [×2], C22×He3, C6×C24, C8×He3 [×2], C2×C4×He3, C2×C8×He3

Smallest permutation representation of C2×C8×He3
On 144 points
Generators in S144
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(33 96)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(57 144)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 112)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)(81 134)(82 135)(83 136)(84 129)(85 130)(86 131)(87 132)(88 133)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 113)
(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 86 143)(2 87 144)(3 88 137)(4 81 138)(5 82 139)(6 83 140)(7 84 141)(8 85 142)(9 131 64)(10 132 57)(11 133 58)(12 134 59)(13 135 60)(14 136 61)(15 129 62)(16 130 63)(17 50 72)(18 51 65)(19 52 66)(20 53 67)(21 54 68)(22 55 69)(23 56 70)(24 49 71)(25 105 46)(26 106 47)(27 107 48)(28 108 41)(29 109 42)(30 110 43)(31 111 44)(32 112 45)(33 113 76)(34 114 77)(35 115 78)(36 116 79)(37 117 80)(38 118 73)(39 119 74)(40 120 75)(89 97 122)(90 98 123)(91 99 124)(92 100 125)(93 101 126)(94 102 127)(95 103 128)(96 104 121)
(1 35 27)(2 36 28)(3 37 29)(4 38 30)(5 39 31)(6 40 32)(7 33 25)(8 34 26)(9 90 54)(10 91 55)(11 92 56)(12 93 49)(13 94 50)(14 95 51)(15 96 52)(16 89 53)(17 60 127)(18 61 128)(19 62 121)(20 63 122)(21 64 123)(22 57 124)(23 58 125)(24 59 126)(41 144 79)(42 137 80)(43 138 73)(44 139 74)(45 140 75)(46 141 76)(47 142 77)(48 143 78)(65 136 103)(66 129 104)(67 130 97)(68 131 98)(69 132 99)(70 133 100)(71 134 101)(72 135 102)(81 118 110)(82 119 111)(83 120 112)(84 113 105)(85 114 106)(86 115 107)(87 116 108)(88 117 109)
(1 143 107)(2 144 108)(3 137 109)(4 138 110)(5 139 111)(6 140 112)(7 141 105)(8 142 106)(9 64 68)(10 57 69)(11 58 70)(12 59 71)(13 60 72)(14 61 65)(15 62 66)(16 63 67)(17 102 50)(18 103 51)(19 104 52)(20 97 53)(21 98 54)(22 99 55)(23 100 56)(24 101 49)(25 46 113)(26 47 114)(27 48 115)(28 41 116)(29 42 117)(30 43 118)(31 44 119)(32 45 120)(33 76 84)(34 77 85)(35 78 86)(36 79 87)(37 80 88)(38 73 81)(39 74 82)(40 75 83)(89 122 130)(90 123 131)(91 124 132)(92 125 133)(93 126 134)(94 127 135)(95 128 136)(96 121 129)

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

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

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

176 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3J 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AD 8A ··· 8H 12A ··· 12H 12I ··· 12AN 24A ··· 24P 24Q ··· 24CB order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 1 1 3 ··· 3 1 1 1 1 1 ··· 1 3 ··· 3 1 ··· 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 He3 C2×He3 C2×He3 C4×He3 C4×He3 C8×He3 kernel C2×C8×He3 C8×He3 C2×C4×He3 C6×C24 C4×He3 C22×He3 C3×C24 C6×C12 C2×He3 C3×C12 C62 C3×C6 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 8 2 2 16 8 8 16 16 64 2 4 2 4 4 16

Matrix representation of C2×C8×He3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 10 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 1 0 0 0 0 0 1 0 0 72 72 7 0 0 0 1
,
 1 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 8 0 0 0 0 65 65 56 0 1 0 0 0 72 8 8
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,0,72,0,0,1,72,0,0,0,7,1],[1,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,65,1,72,0,65,0,8,0,56,0,8] >;

C2×C8×He3 in GAP, Magma, Sage, TeX

C_2\times C_8\times {\rm He}_3
% in TeX

G:=Group("C2xC8xHe3");
// GroupNames label

G:=SmallGroup(432,210);
// by ID

G=gap.SmallGroup(432,210);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,605,242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^3=d^3=e^3=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^-1=c*d^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽