Copied to
clipboard

## G = C9×C4.10D4order 288 = 25·32

### Direct product of C9 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×C4.10D4
 Chief series C1 — C2 — C6 — C12 — C2×C12 — C2×C36 — C9×M4(2) — C9×C4.10D4
 Lower central C1 — C2 — C22 — C9×C4.10D4
 Upper central C1 — C18 — C2×C36 — C9×C4.10D4

Generators and relations for C9×C4.10D4
G = < a,b,c,d | a9=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 78 in 57 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C9, C12, C12, C2×C6, M4(2), C2×Q8, C18, C18, C24, C2×C12, C2×C12, C3×Q8, C4.10D4, C36, C36, C2×C18, C3×M4(2), C6×Q8, C72, C2×C36, C2×C36, Q8×C9, C3×C4.10D4, C9×M4(2), Q8×C18, C9×C4.10D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C4.10D4, C36, C2×C18, C3×C22⋊C4, C2×C36, D4×C9, C3×C4.10D4, C9×C22⋊C4, C9×C4.10D4

Smallest permutation representation of C9×C4.10D4
On 144 points
Generators in S144
(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 59 42 53)(2 60 43 54)(3 61 44 46)(4 62 45 47)(5 63 37 48)(6 55 38 49)(7 56 39 50)(8 57 40 51)(9 58 41 52)(10 142 25 32)(11 143 26 33)(12 144 27 34)(13 136 19 35)(14 137 20 36)(15 138 21 28)(16 139 22 29)(17 140 23 30)(18 141 24 31)(64 76 93 87)(65 77 94 88)(66 78 95 89)(67 79 96 90)(68 80 97 82)(69 81 98 83)(70 73 99 84)(71 74 91 85)(72 75 92 86)(100 112 129 123)(101 113 130 124)(102 114 131 125)(103 115 132 126)(104 116 133 118)(105 117 134 119)(106 109 135 120)(107 110 127 121)(108 111 128 122)
(1 131 59 114 42 102 53 125)(2 132 60 115 43 103 54 126)(3 133 61 116 44 104 46 118)(4 134 62 117 45 105 47 119)(5 135 63 109 37 106 48 120)(6 127 55 110 38 107 49 121)(7 128 56 111 39 108 50 122)(8 129 57 112 40 100 51 123)(9 130 58 113 41 101 52 124)(10 73 32 99 25 84 142 70)(11 74 33 91 26 85 143 71)(12 75 34 92 27 86 144 72)(13 76 35 93 19 87 136 64)(14 77 36 94 20 88 137 65)(15 78 28 95 21 89 138 66)(16 79 29 96 22 90 139 67)(17 80 30 97 23 82 140 68)(18 81 31 98 24 83 141 69)
(1 138 53 15 42 28 59 21)(2 139 54 16 43 29 60 22)(3 140 46 17 44 30 61 23)(4 141 47 18 45 31 62 24)(5 142 48 10 37 32 63 25)(6 143 49 11 38 33 55 26)(7 144 50 12 39 34 56 27)(8 136 51 13 40 35 57 19)(9 137 52 14 41 36 58 20)(64 112 87 100 93 123 76 129)(65 113 88 101 94 124 77 130)(66 114 89 102 95 125 78 131)(67 115 90 103 96 126 79 132)(68 116 82 104 97 118 80 133)(69 117 83 105 98 119 81 134)(70 109 84 106 99 120 73 135)(71 110 85 107 91 121 74 127)(72 111 86 108 92 122 75 128)

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

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

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

99 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 18G ··· 18L 24A ··· 24H 36A ··· 36L 36M ··· 36X 72A ··· 72X order 1 2 2 3 3 4 4 4 4 6 6 6 6 8 8 8 8 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 1 1 2 2 4 4 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 D4 C3×D4 D4×C9 C4.10D4 C3×C4.10D4 C9×C4.10D4 kernel C9×C4.10D4 C9×M4(2) Q8×C18 C3×C4.10D4 C2×C36 C3×M4(2) C6×Q8 C4.10D4 C2×C12 M4(2) C2×Q8 C2×C4 C36 C12 C4 C9 C3 C1 # reps 1 2 1 2 4 4 2 6 8 12 6 24 2 4 12 1 2 6

Matrix representation of C9×C4.10D4 in GL4(𝔽73) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 17 1 6 28 2 56 16 41 0 0 0 72 0 0 1 0
,
 32 12 0 44 0 0 0 1 0 1 0 0 2 56 16 41
,
 55 23 70 27 0 0 43 62 13 10 31 11 51 11 43 60
G:=sub<GL(4,GF(73))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[17,2,0,0,1,56,0,0,6,16,0,1,28,41,72,0],[32,0,0,2,12,0,1,56,0,0,0,16,44,1,0,41],[55,0,13,51,23,0,10,11,70,43,31,43,27,62,11,60] >;

C9×C4.10D4 in GAP, Magma, Sage, TeX

C_9\times C_4._{10}D_4
% in TeX

G:=Group("C9xC4.10D4");
// GroupNames label

G:=SmallGroup(288,51);
// by ID

G=gap.SmallGroup(288,51);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,1016,268,4371,2951,242]);
// Polycyclic

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

׿
×
𝔽