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G = C24×He3  order 432 = 24·33

Direct product of C24 and He3

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

Aliases: C24×He3, (C2×C62)⋊8C6, C62⋊15(C2×C6), (C2×C6).39C62, C6.16(C2×C62), C32⋊3(C23×C6), (C22×C62)⋊3C3, C3.1(C22×C62), (C23×C6).16C32, (C3×C6)⋊3(C22×C6), (C22×C6).24(C3×C6), SmallGroup(432,563)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C24×He3
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C23×He3 — C24×He3
Lower central
C1 — C3 — C24×He3
Upper central
C1 — C23×C6 — C24×He3

Generators and relations for C24×He3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e3=f3=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, geg-1=ef-1, fg=gf >

Subgroups: 1273 in 737 conjugacy classes, 469 normal (6 characteristic)
C1, C2, C3, C3, C22, C6, C6, C23, C32, C2×C6, C2×C6, C24, C3×C6, C22×C6, C22×C6, He3, C62, C23×C6, C23×C6, C2×He3, C2×C62, C22×He3, C22×C62, C23×He3, C24×He3
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C24, C3×C6, C22×C6, He3, C62, C23×C6, C2×He3, C2×C62, C22×He3, C22×C62, C23×He3, C24×He3

Smallest permutation representation of C24×He3
►On 144 points
Generators in S144
(1 31)(2 17)(3 19)(4 7)(5 12)(6 8)(9 24)(10 45)(11 44)(13 37)(14 29)(15 30)(16 43)(18 27)(20 41)(21 42)(22 26)(23 25)(28 40)(32 47)(33 48)(34 46)(35 39)(36 38)(49 71)(50 72)(51 70)(52 105)(53 103)(54 104)(55 79)(56 80)(57 81)(58 73)(59 74)(60 75)(61 82)(62 83)(63 84)(64 142)(65 143)(66 144)(67 107)(68 108)(69 106)(76 100)(77 101)(78 102)(85 132)(86 130)(87 131)(88 98)(89 99)(90 97)(91 115)(92 116)(93 117)(94 114)(95 112)(96 113)(109 138)(110 136)(111 137)(118 121)(119 122)(120 123)(124 134)(125 135)(126 133)(127 141)(128 139)(129 140)

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

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

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

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

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

176 conjugacy classes

class 1 2A···2O3A3B3C···3J6A···6AD6AE···6ET
order12···2333···36···66···6
size11···1113···31···13···3

176 irreducible representations

dim111133
type++
imageC1C2C3C6He3C2×He3
kernelC24×He3C23×He3C22×C62C2×C62C24C23
# reps1158120230

Matrix representation of C24×He3 ►in GL7(𝔽7)

1000000
0100000
0060000
0006000
0000100
0000010
0000001
,
6000000
0600000
0010000
0006000
0000100
0000010
0000001
,
6000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0060000
0001000
0000100
0000010
0000001
,
4000000
0400000
0040000
0002000
0000100
0000040
0000232
,
1000000
0100000
0010000
0001000
0000400
0000040
0000004
,
4000000
0200000
0040000
0004000
0000010
0000663
0000001

G:=sub<GL(7,GF(7))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,4,3,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,3,1] >;
 

C24×He3 in GAP, Magma, Sage, TeX 

C_2^4\times {\rm He}_3
 
% in TeX
 
G:=Group("C2^4xHe3");
 
// GroupNames label
 
G:=SmallGroup(432,563);
 
// by ID
 
G=gap.SmallGroup(432,563);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,537]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^3=f^3=g^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,g*e*g^-1=e*f^-1,f*g=g*f>;
 
// generators/relations
 

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