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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

Aliases: C2×C8×3- 1+2, C728C6, C182C24, C36.6C12, C62.10C12, C12.39C62, (C2×C72)⋊C3, (C6×C24).C3, C93(C2×C24), C3.2(C6×C24), (C3×C6).6C24, C6.5(C3×C24), C32.(C2×C24), (C3×C24).10C6, C24.19(C3×C6), (C2×C36).11C6, (C6×C12).19C6, C6.15(C6×C12), (C2×C18).5C12, C36.28(C2×C6), (C3×C12).14C12, C18.13(C2×C12), C12.16(C3×C12), (C2×C24).2C32, C4.3(C4×3- 1+2), (C4×3- 1+2).6C4, C22.2(C4×3- 1+2), C4.5(C22×3- 1+2), (C22×3- 1+2).4C4, (C4×3- 1+2).21C22, (C3×C12).67(C2×C6), (C3×C6).33(C2×C12), (C2×C12).31(C3×C6), (C2×C6).16(C3×C12), C2.2(C2×C4×3- 1+2), (C2×C4×3- 1+2).11C2, (C2×C4).5(C2×3- 1+2), (C2×3- 1+2).13(C2×C4), SmallGroup(432,211)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C8×3- 1+2
C1C2C6C12C3×C12C4×3- 1+2C8×3- 1+2 — C2×C8×3- 1+2
C1C3 — C2×C8×3- 1+2
C1C2×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 2A2B2C3A3B3C3D4A4B4C4D6A···6F6G···6L8A···8H9A···9F12A···12H12I···12P18A···18R24A···24P24Q···24AF36A···36X72A···72AV
order1222333344446···66···68···89···912···1212···1218···1824···2424···2436···3672···72
size1111113311111···13···31···13···31···13···33···31···13···33···33···3

176 irreducible representations

dim111111111111111111333333
type+++
imageC1C2C2C3C3C4C4C6C6C6C6C8C12C12C12C12C24C243- 1+2C2×3- 1+2C2×3- 1+2C4×3- 1+2C4×3- 1+2C8×3- 1+2
kernelC2×C8×3- 1+2C8×3- 1+2C2×C4×3- 1+2C2×C72C6×C24C4×3- 1+2C22×3- 1+2C72C2×C36C3×C24C6×C12C2×3- 1+2C36C2×C18C3×C12C62C18C3×C6C2×C8C8C2×C4C4C22C2
# reps121622212642812124448162424416

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

72000
07200
00720
00072
,
72000
01000
00100
00010
,
64000
01710
033728
00720
,
64000
0100
03380
05064
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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