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## G = C2×C8×3- 1+2order 432 = 24·33

### Direct product of C2×C8 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C8×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — C8×3- 1+2 — C2×C8×3- 1+2
 Lower central C1 — C3 — C2×C8×3- 1+2
 Upper central C1 — C2×C24 — C2×C8×3- 1+2

Generators and relations for C2×C8×3- 1+2
G = < a,b,c,d | a2=b8=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 110 in 88 conjugacy classes, 77 normal (28 characteristic)
C1, C2, C2 [×2], C3, C3, C4 [×2], C22, C6, C6 [×2], C6 [×3], C8 [×2], C2×C4, C9 [×3], C32, C12 [×2], C12 [×2], C2×C6, C2×C6, C2×C8, C18 [×9], C3×C6, C3×C6 [×2], C24 [×2], C24 [×2], C2×C12, C2×C12, 3- 1+2, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C2×C24, C2×C24, C2×3- 1+2, C2×3- 1+2 [×2], C72 [×6], C2×C36 [×3], C3×C24 [×2], C6×C12, C4×3- 1+2 [×2], C22×3- 1+2, C2×C72 [×3], C6×C24, C8×3- 1+2 [×2], C2×C4×3- 1+2, C2×C8×3- 1+2
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], 3- 1+2, C3×C12 [×2], C62, C2×C24 [×4], C2×3- 1+2 [×3], C3×C24 [×2], C6×C12, C4×3- 1+2 [×2], C22×3- 1+2, C6×C24, C8×3- 1+2 [×2], C2×C4×3- 1+2, C2×C8×3- 1+2

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

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

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

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

176 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6F 6G ··· 6L 8A ··· 8H 9A ··· 9F 12A ··· 12H 12I ··· 12P 18A ··· 18R 24A ··· 24P 24Q ··· 24AF 36A ··· 36X 72A ··· 72AV order 1 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 24 ··· 24 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 1 1 3 3 1 1 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + image C1 C2 C2 C3 C3 C4 C4 C6 C6 C6 C6 C8 C12 C12 C12 C12 C24 C24 3- 1+2 C2×3- 1+2 C2×3- 1+2 C4×3- 1+2 C4×3- 1+2 C8×3- 1+2 kernel C2×C8×3- 1+2 C8×3- 1+2 C2×C4×3- 1+2 C2×C72 C6×C24 C4×3- 1+2 C22×3- 1+2 C72 C2×C36 C3×C24 C6×C12 C2×3- 1+2 C36 C2×C18 C3×C12 C62 C18 C3×C6 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 6 2 2 2 12 6 4 2 8 12 12 4 4 48 16 2 4 2 4 4 16

Matrix representation of C2×C8×3- 1+2 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 64 0 0 0 0 1 71 0 0 33 72 8 0 0 72 0
,
 64 0 0 0 0 1 0 0 0 33 8 0 0 5 0 64
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[64,0,0,0,0,1,33,0,0,71,72,72,0,0,8,0],[64,0,0,0,0,1,33,5,0,0,8,0,0,0,0,64] >;

C2×C8×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times C_8\times 3_-^{1+2}
% in TeX

G:=Group("C2xC8xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,211);
// by ID

G=gap.SmallGroup(432,211);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,394,605,242]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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