Copied to
clipboard

G = C9×C4.4D4order 288 = 25·32

Direct product of C9 and C4.4D4

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

Aliases: C9×C4.4D4, C428C18, C36.40D4, C4.4(D4×C9), (C4×C36)⋊12C2, (C2×Q8)⋊4C18, (Q8×C18)⋊9C2, C2.8(D4×C18), C6.71(C6×D4), C22⋊C45C18, (C4×C12).18C6, (C2×D4).5C18, (C6×D4).12C6, C18.71(C2×D4), C12.39(C3×D4), (C6×Q8).17C6, (D4×C18).12C2, C23.4(C2×C18), C18.44(C4○D4), (C2×C18).79C23, (C2×C36).80C22, (C22×C18).2C22, C22.14(C22×C18), C2.7(C9×C4○D4), C3.(C3×C4.4D4), C6.44(C3×C4○D4), (C3×C4.4D4).C3, (C9×C22⋊C4)⋊13C2, (C2×C12).83(C2×C6), (C2×C4).20(C2×C18), (C3×C22⋊C4).9C6, (C22×C6).7(C2×C6), (C2×C6).84(C22×C6), SmallGroup(288,174)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C4.4D4
C1C3C6C2×C6C2×C18C22×C18C9×C22⋊C4 — C9×C4.4D4
C1C22 — C9×C4.4D4
C1C2×C18 — C9×C4.4D4

Generators and relations for C9×C4.4D4
 G = < a,b,c,d | a9=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 174 in 114 conjugacy classes, 66 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C9, C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C4.4D4, C36 [×2], C36 [×4], C2×C18, C2×C18 [×6], C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C2×C36, C2×C36 [×4], D4×C9 [×2], Q8×C9 [×2], C22×C18 [×2], C3×C4.4D4, C4×C36, C9×C22⋊C4 [×4], D4×C18, Q8×C18, C9×C4.4D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C4○D4 [×2], C18 [×7], C3×D4 [×2], C22×C6, C4.4D4, C2×C18 [×7], C6×D4, C3×C4○D4 [×2], D4×C9 [×2], C22×C18, C3×C4.4D4, D4×C18, C9×C4○D4 [×2], C9×C4.4D4

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

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4F4G4H6A···6F6G6H6I6J9A···9F12A···12L12M12N12O12P18A···18R18S···18AD36A···36AJ36AK···36AV
order122222334···4446···666669···912···121212121218···1818···1836···3636···36
size111144112···2441···144441···12···244441···14···42···24···4

126 irreducible representations

dim111111111111111222222
type++++++
imageC1C2C2C2C2C3C6C6C6C6C9C18C18C18C18D4C4○D4C3×D4C3×C4○D4D4×C9C9×C4○D4
kernelC9×C4.4D4C4×C36C9×C22⋊C4D4×C18Q8×C18C3×C4.4D4C4×C12C3×C22⋊C4C6×D4C6×Q8C4.4D4C42C22⋊C4C2×D4C2×Q8C36C18C12C6C4C2
# reps114112282266246624481224

Matrix representation of C9×C4.4D4 in GL4(𝔽37) generated by

12000
01200
0010
0001
,
36000
03600
0006
0060
,
22100
353500
00310
00031
,
22100
213500
00310
0006
G:=sub<GL(4,GF(37))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,0,6,0,0,6,0],[2,35,0,0,21,35,0,0,0,0,31,0,0,0,0,31],[2,21,0,0,21,35,0,0,0,0,31,0,0,0,0,6] >;

C9×C4.4D4 in GAP, Magma, Sage, TeX

C_9\times C_4._4D_4
% in TeX

G:=Group("C9xC4.4D4");
// GroupNames label

G:=SmallGroup(288,174);
// by ID

G=gap.SmallGroup(288,174);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,344,1094,142,360]);
// Polycyclic

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

׿
×
𝔽