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## G = C2×Q8⋊3- 1+2order 432 = 24·33

### Direct product of C2 and Q8⋊3- 1+2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C2×Q8⋊3- 1+2
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — Q8⋊3- 1+2 — C2×Q8⋊3- 1+2
 Lower central Q8 — C3×Q8 — C2×Q8⋊3- 1+2
 Upper central C1 — C2×C6 — C62

Generators and relations for C2×Q8⋊3- 1+2
G = < a,b,c,d,e | a2=b4=d9=e3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, be=eb, dcd-1=bc, ce=ec, ede-1=d4 >

Subgroups: 202 in 80 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C3×Q8, C3×Q8, 3- 1+2, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C2×3- 1+2, Q8⋊C9, C6×C12, Q8×C32, Q8×C32, C22×3- 1+2, C2×Q8⋊C9, Q8×C3×C6, Q8⋊3- 1+2, C2×Q8⋊3- 1+2
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, SL2(𝔽3), C2×A4, 3- 1+2, C3×A4, C2×SL2(𝔽3), C2×3- 1+2, C3×SL2(𝔽3), C6×A4, C32.A4, C6×SL2(𝔽3), Q8⋊3- 1+2, C2×C32.A4, C2×Q8⋊3- 1+2

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 6A ··· 6F 6G ··· 6L 9A ··· 9F 12A ··· 12P 18A ··· 18R order 1 2 2 2 3 3 3 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 1 1 3 3 6 6 1 ··· 1 3 ··· 3 12 ··· 12 6 ··· 6 12 ··· 12

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 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C32.A4 C2×C32.A4 Q8⋊3- 1+2 kernel C2×Q8⋊3- 1+2 Q8⋊3- 1+2 C2×Q8⋊C9 Q8×C3×C6 Q8⋊C9 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⋊3- 1+2 in GL7(𝔽37)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 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 6 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 31 0 0 0 0 0 31 0 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 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 28 9 0 0 0 0 0 20 20 0 0 0 0 0 0 0 20 17 0 0 0 0 0 28 28 0 0 0 0 0 0 0 21 1 0 0 0 0 0 30 15 21 0 0 0 0 25 10 1
,
 26 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 7 26 0 0 0 0 0 26 36 10

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,6,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[28,20,0,0,0,0,0,9,20,0,0,0,0,0,0,0,20,28,0,0,0,0,0,17,28,0,0,0,0,0,0,0,21,30,25,0,0,0,0,1,15,10,0,0,0,0,0,21,1],[26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,7,26,0,0,0,0,0,26,36,0,0,0,0,0,0,10] >;

C2×Q8⋊3- 1+2 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes 3_-^{1+2}
% in TeX

G:=Group("C2xQ8:ES-(3,1)");
// GroupNames label

G:=SmallGroup(432,335);
// by ID

G=gap.SmallGroup(432,335);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,134,261,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=c,b*e=e*b,d*c*d^-1=b*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

׿
×
𝔽