Copied to
clipboard

## G = C2×Q8⋊He3order 432 = 24·33

### Direct product of C2 and Q8⋊He3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C2×Q8⋊He3
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — Q8⋊He3 — C2×Q8⋊He3
 Lower central Q8 — C3×Q8 — C2×Q8⋊He3
 Upper central C1 — C2×C6 — C62

Generators and relations for C2×Q8⋊He3
G = < a,b,c,d,e,f | a2=b4=d3=e3=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=c, cd=dc, ce=ec, fcf-1=bc, de=ed, fdf-1=de-1, ef=fe >

Subgroups: 400 in 101 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C32, C32, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, He3, C3×C12, C62, C62, C2×SL2(𝔽3), C6×Q8, C6×Q8, C2×He3, C3×SL2(𝔽3), C6×C12, Q8×C32, Q8×C32, C22×He3, C6×SL2(𝔽3), Q8×C3×C6, Q8⋊He3, C2×Q8⋊He3
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, SL2(𝔽3), C2×A4, He3, C3×A4, C2×SL2(𝔽3), C2×He3, C3×SL2(𝔽3), C6×A4, C32⋊A4, C6×SL2(𝔽3), Q8⋊He3, C2×C32⋊A4, C2×Q8⋊He3

Smallest permutation representation of C2×Q8⋊He3
On 144 points
Generators in S144
(1 82)(2 83)(3 84)(4 81)(5 75)(6 76)(7 73)(8 74)(9 79)(10 80)(11 77)(12 78)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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)
(1 12 3 10)(2 11 4 9)(5 144 7 142)(6 143 8 141)(13 17 15 19)(14 20 16 18)(21 27 23 25)(22 26 24 28)(29 36 31 34)(30 35 32 33)(37 44 39 42)(38 43 40 41)(45 49 47 51)(46 52 48 50)(53 59 55 57)(54 58 56 60)(61 67 63 65)(62 66 64 68)(69 76 71 74)(70 75 72 73)(77 81 79 83)(78 84 80 82)(85 89 87 91)(86 92 88 90)(93 99 95 97)(94 98 96 100)(101 108 103 106)(102 107 104 105)(109 116 111 114)(110 115 112 113)(117 121 119 123)(118 124 120 122)(125 131 127 129)(126 130 128 132)(133 139 135 137)(134 138 136 140)
(5 130 138)(6 131 139)(7 132 140)(8 129 137)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(57 65 74)(58 66 75)(59 67 76)(60 68 73)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 122 113)(106 123 114)(107 124 115)(108 121 116)(125 133 141)(126 134 142)(127 135 143)(128 136 144)
(1 28 18)(2 25 19)(3 26 20)(4 27 17)(5 138 130)(6 139 131)(7 140 132)(8 137 129)(9 23 15)(10 24 16)(11 21 13)(12 22 14)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 50 41)(34 51 42)(35 52 43)(36 49 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 74 65)(58 75 66)(59 76 67)(60 73 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(81 99 89)(82 100 90)(83 97 91)(84 98 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 122 113)(106 123 114)(107 124 115)(108 121 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(1 53 32)(2 58 33)(3 55 30)(4 60 35)(5 122 97)(6 117 93)(7 124 99)(8 119 95)(9 57 31)(10 56 34)(11 59 29)(12 54 36)(13 67 37)(14 62 44)(15 65 39)(16 64 42)(17 68 43)(18 61 40)(19 66 41)(20 63 38)(21 76 45)(22 70 49)(23 74 47)(24 72 51)(25 75 50)(26 71 46)(27 73 52)(28 69 48)(77 131 101)(78 126 108)(79 129 103)(80 128 106)(81 132 107)(82 125 104)(83 130 105)(84 127 102)(85 139 109)(86 134 116)(87 137 111)(88 136 114)(89 140 115)(90 133 112)(91 138 113)(92 135 110)(94 142 121)(96 144 123)(98 143 118)(100 141 120)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E ··· 3J 4A 4B 6A ··· 6F 6G ··· 6L 6M ··· 6AD 12A ··· 12P order 1 2 2 2 3 3 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 1 1 3 3 12 ··· 12 6 6 1 ··· 1 3 ··· 3 12 ··· 12 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 3 6 type + + - + + image C1 C2 C3 C3 C6 C6 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 C2×A4 He3 C3×A4 C2×He3 C6×A4 C32⋊A4 C2×C32⋊A4 Q8⋊He3 kernel C2×Q8⋊He3 Q8⋊He3 C6×SL2(𝔽3) Q8×C3×C6 C3×SL2(𝔽3) Q8×C32 C3×C6 C3×C6 C6 C62 C3×C6 C2×Q8 C2×C6 Q8 C6 C22 C2 C2 # reps 1 1 6 2 6 2 2 4 12 1 1 2 2 2 2 6 6 4

Matrix representation of C2×Q8⋊He3 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 3 4 0 0 0 4 10 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12
,
 0 1 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 10 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,4,0,0,0,4,10,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[0,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,10,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C2×Q8⋊He3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes {\rm He}_3
% in TeX

G:=Group("C2xQ8:He3");
// GroupNames label

G:=SmallGroup(432,336);
// by ID

G=gap.SmallGroup(432,336);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,261,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=d^3=e^3=f^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=c,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,d*e=e*d,f*d*f^-1=d*e^-1,e*f=f*e>;
// generators/relations

׿
×
𝔽