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G = C9×C4.10D4order 288 = 25·32

Direct product of C9 and C4.10D4

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

Aliases: C9×C4.10D4, C36.59D4, M4(2).1C18, (C2×C4).C36, (C2×C36).2C4, C4.10(D4×C9), (C2×C12).2C12, C12.68(C3×D4), (C2×Q8).3C18, (Q8×C18).6C2, (C6×Q8).14C6, C22.4(C2×C36), (C2×C36).59C22, (C9×M4(2)).3C2, (C3×M4(2)).4C6, C18.23(C22⋊C4), (C2×C4).2(C2×C18), C2.5(C9×C22⋊C4), C3.(C3×C4.10D4), (C2×C6).25(C2×C12), (C2×C12).61(C2×C6), (C2×C18).21(C2×C4), C6.23(C3×C22⋊C4), (C3×C4.10D4).C3, SmallGroup(288,51)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C4.10D4
C1C2C6C12C2×C12C2×C36C9×M4(2) — C9×C4.10D4
C1C2C22 — C9×C4.10D4
C1C18C2×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 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, C12 [×2], C12 [×2], C2×C6, M4(2) [×2], C2×Q8, C18, C18, C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C4.10D4, C36 [×2], C36 [×2], C2×C18, C3×M4(2) [×2], C6×Q8, C72 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C3×C4.10D4, C9×M4(2) [×2], Q8×C18, C9×C4.10D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, C18 [×3], C2×C12, C3×D4 [×2], C4.10D4, C36 [×2], C2×C18, C3×C22⋊C4, C2×C36, D4×C9 [×2], 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 47)(2 60 43 48)(3 61 44 49)(4 62 45 50)(5 63 37 51)(6 55 38 52)(7 56 39 53)(8 57 40 54)(9 58 41 46)(10 34 22 144)(11 35 23 136)(12 36 24 137)(13 28 25 138)(14 29 26 139)(15 30 27 140)(16 31 19 141)(17 32 20 142)(18 33 21 143)(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 47 125)(2 132 60 115 43 103 48 126)(3 133 61 116 44 104 49 118)(4 134 62 117 45 105 50 119)(5 135 63 109 37 106 51 120)(6 127 55 110 38 107 52 121)(7 128 56 111 39 108 53 122)(8 129 57 112 40 100 54 123)(9 130 58 113 41 101 46 124)(10 86 144 72 22 75 34 92)(11 87 136 64 23 76 35 93)(12 88 137 65 24 77 36 94)(13 89 138 66 25 78 28 95)(14 90 139 67 26 79 29 96)(15 82 140 68 27 80 30 97)(16 83 141 69 19 81 31 98)(17 84 142 70 20 73 32 99)(18 85 143 71 21 74 33 91)
(1 138 47 25 42 28 59 13)(2 139 48 26 43 29 60 14)(3 140 49 27 44 30 61 15)(4 141 50 19 45 31 62 16)(5 142 51 20 37 32 63 17)(6 143 52 21 38 33 55 18)(7 144 53 22 39 34 56 10)(8 136 54 23 40 35 57 11)(9 137 46 24 41 36 58 12)(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), (1,59,42,47)(2,60,43,48)(3,61,44,49)(4,62,45,50)(5,63,37,51)(6,55,38,52)(7,56,39,53)(8,57,40,54)(9,58,41,46)(10,34,22,144)(11,35,23,136)(12,36,24,137)(13,28,25,138)(14,29,26,139)(15,30,27,140)(16,31,19,141)(17,32,20,142)(18,33,21,143)(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), (1,131,59,114,42,102,47,125)(2,132,60,115,43,103,48,126)(3,133,61,116,44,104,49,118)(4,134,62,117,45,105,50,119)(5,135,63,109,37,106,51,120)(6,127,55,110,38,107,52,121)(7,128,56,111,39,108,53,122)(8,129,57,112,40,100,54,123)(9,130,58,113,41,101,46,124)(10,86,144,72,22,75,34,92)(11,87,136,64,23,76,35,93)(12,88,137,65,24,77,36,94)(13,89,138,66,25,78,28,95)(14,90,139,67,26,79,29,96)(15,82,140,68,27,80,30,97)(16,83,141,69,19,81,31,98)(17,84,142,70,20,73,32,99)(18,85,143,71,21,74,33,91), (1,138,47,25,42,28,59,13)(2,139,48,26,43,29,60,14)(3,140,49,27,44,30,61,15)(4,141,50,19,45,31,62,16)(5,142,51,20,37,32,63,17)(6,143,52,21,38,33,55,18)(7,144,53,22,39,34,56,10)(8,136,54,23,40,35,57,11)(9,137,46,24,41,36,58,12)(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), (1,59,42,47)(2,60,43,48)(3,61,44,49)(4,62,45,50)(5,63,37,51)(6,55,38,52)(7,56,39,53)(8,57,40,54)(9,58,41,46)(10,34,22,144)(11,35,23,136)(12,36,24,137)(13,28,25,138)(14,29,26,139)(15,30,27,140)(16,31,19,141)(17,32,20,142)(18,33,21,143)(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), (1,131,59,114,42,102,47,125)(2,132,60,115,43,103,48,126)(3,133,61,116,44,104,49,118)(4,134,62,117,45,105,50,119)(5,135,63,109,37,106,51,120)(6,127,55,110,38,107,52,121)(7,128,56,111,39,108,53,122)(8,129,57,112,40,100,54,123)(9,130,58,113,41,101,46,124)(10,86,144,72,22,75,34,92)(11,87,136,64,23,76,35,93)(12,88,137,65,24,77,36,94)(13,89,138,66,25,78,28,95)(14,90,139,67,26,79,29,96)(15,82,140,68,27,80,30,97)(16,83,141,69,19,81,31,98)(17,84,142,70,20,73,32,99)(18,85,143,71,21,74,33,91), (1,138,47,25,42,28,59,13)(2,139,48,26,43,29,60,14)(3,140,49,27,44,30,61,15)(4,141,50,19,45,31,62,16)(5,142,51,20,37,32,63,17)(6,143,52,21,38,33,55,18)(7,144,53,22,39,34,56,10)(8,136,54,23,40,35,57,11)(9,137,46,24,41,36,58,12)(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)], [(1,59,42,47),(2,60,43,48),(3,61,44,49),(4,62,45,50),(5,63,37,51),(6,55,38,52),(7,56,39,53),(8,57,40,54),(9,58,41,46),(10,34,22,144),(11,35,23,136),(12,36,24,137),(13,28,25,138),(14,29,26,139),(15,30,27,140),(16,31,19,141),(17,32,20,142),(18,33,21,143),(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)], [(1,131,59,114,42,102,47,125),(2,132,60,115,43,103,48,126),(3,133,61,116,44,104,49,118),(4,134,62,117,45,105,50,119),(5,135,63,109,37,106,51,120),(6,127,55,110,38,107,52,121),(7,128,56,111,39,108,53,122),(8,129,57,112,40,100,54,123),(9,130,58,113,41,101,46,124),(10,86,144,72,22,75,34,92),(11,87,136,64,23,76,35,93),(12,88,137,65,24,77,36,94),(13,89,138,66,25,78,28,95),(14,90,139,67,26,79,29,96),(15,82,140,68,27,80,30,97),(16,83,141,69,19,81,31,98),(17,84,142,70,20,73,32,99),(18,85,143,71,21,74,33,91)], [(1,138,47,25,42,28,59,13),(2,139,48,26,43,29,60,14),(3,140,49,27,44,30,61,15),(4,141,50,19,45,31,62,16),(5,142,51,20,37,32,63,17),(6,143,52,21,38,33,55,18),(7,144,53,22,39,34,56,10),(8,136,54,23,40,35,57,11),(9,137,46,24,41,36,58,12),(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 2A2B3A3B4A4B4C4D6A6B6C6D8A8B8C8D9A···9F12A12B12C12D12E12F12G12H18A···18F18G···18L24A···24H36A···36L36M···36X72A···72X
order122334444666688889···9121212121212121218···1818···1824···2436···3636···3672···72
size112112244112244441···1222244441···12···24···42···24···44···4

99 irreducible representations

dim111111111111222444
type++++-
imageC1C2C2C3C4C6C6C9C12C18C18C36D4C3×D4D4×C9C4.10D4C3×C4.10D4C9×C4.10D4
kernelC9×C4.10D4C9×M4(2)Q8×C18C3×C4.10D4C2×C36C3×M4(2)C6×Q8C4.10D4C2×C12M4(2)C2×Q8C2×C4C36C12C4C9C3C1
# reps121244268126242412126

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

37000
03700
00370
00037
,
171628
2561641
00072
0010
,
3212044
0001
0100
2561641
,
55237027
004362
13103111
51114360
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

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