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G = C4.D36order 288 = 25·32

3rd non-split extension by C4 of D36 acting via D36/D18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.3D12, C36.47D4, C4.11D36, M4(2).2D9, (C2×C4).1D18, (C2×Dic9).C4, (C2×C12).42D6, C22.4(C4×D9), C6.15(D6⋊C4), C2.9(D18⋊C4), C4.21(C9⋊D4), C91(C4.10D4), C4.Dic9.3C2, C18.8(C22⋊C4), (C2×C36).20C22, C3.(C12.47D4), (C2×Dic18).6C2, (C9×M4(2)).2C2, (C3×M4(2)).6S3, C12.108(C3⋊D4), (C2×C6).3(C4×S3), (C2×C18).2(C2×C4), SmallGroup(288,30)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C4.D36
C1C3C9C18C36C2×C36C2×Dic18 — C4.D36
C9C18C2×C18 — C4.D36
C1C2C2×C4M4(2)

Generators and relations for C4.D36
 G = < a,b,c | a36=1, b4=c2=a18, bab-1=cac-1=a-1, cbc-1=a9b3 >

Subgroups: 228 in 57 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, Dic3 [×2], C12 [×2], C2×C6, M4(2), M4(2), C2×Q8, C18, C18, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C4.10D4, Dic9 [×2], C36 [×2], C2×C18, C4.Dic3, C3×M4(2), C2×Dic6, C9⋊C8, C72, Dic18 [×2], C2×Dic9 [×2], C2×C36, C12.47D4, C4.Dic9, C9×M4(2), C2×Dic18, C4.D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.10D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.47D4, D18⋊C4, C4.D36

Smallest permutation representation of C4.D36
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)
(1 127 10 118 19 109 28 136)(2 126 11 117 20 144 29 135)(3 125 12 116 21 143 30 134)(4 124 13 115 22 142 31 133)(5 123 14 114 23 141 32 132)(6 122 15 113 24 140 33 131)(7 121 16 112 25 139 34 130)(8 120 17 111 26 138 35 129)(9 119 18 110 27 137 36 128)(37 90 64 99 55 108 46 81)(38 89 65 98 56 107 47 80)(39 88 66 97 57 106 48 79)(40 87 67 96 58 105 49 78)(41 86 68 95 59 104 50 77)(42 85 69 94 60 103 51 76)(43 84 70 93 61 102 52 75)(44 83 71 92 62 101 53 74)(45 82 72 91 63 100 54 73)
(1 90 19 108)(2 89 20 107)(3 88 21 106)(4 87 22 105)(5 86 23 104)(6 85 24 103)(7 84 25 102)(8 83 26 101)(9 82 27 100)(10 81 28 99)(11 80 29 98)(12 79 30 97)(13 78 31 96)(14 77 32 95)(15 76 33 94)(16 75 34 93)(17 74 35 92)(18 73 36 91)(37 118 55 136)(38 117 56 135)(39 116 57 134)(40 115 58 133)(41 114 59 132)(42 113 60 131)(43 112 61 130)(44 111 62 129)(45 110 63 128)(46 109 64 127)(47 144 65 126)(48 143 66 125)(49 142 67 124)(50 141 68 123)(51 140 69 122)(52 139 70 121)(53 138 71 120)(54 137 72 119)

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

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

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

51 conjugacy classes

class 1 2A2B 3 4A4B4C4D6A6B8A8B8C8D9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122344446688889991212121818181818182424242436···3636363672···72
size11222236362444363622222422244444442···24444···4

51 irreducible representations

dim1111122222222222444
type+++++++++++---
imageC1C2C2C2C4S3D4D6D9D12C3⋊D4C4×S3D18D36C9⋊D4C4×D9C4.10D4C12.47D4C4.D36
kernelC4.D36C4.Dic9C9×M4(2)C2×Dic18C2×Dic9C3×M4(2)C36C2×C12M4(2)C12C12C2×C6C2×C4C4C4C22C9C3C1
# reps1111412132223666126

Matrix representation of C4.D36 in GL4(𝔽73) generated by

251900
544400
002519
005444
,
00145
001959
511200
632200
,
14500
195900
00145
001959
G:=sub<GL(4,GF(73))| [25,54,0,0,19,44,0,0,0,0,25,54,0,0,19,44],[0,0,51,63,0,0,12,22,14,19,0,0,5,59,0,0],[14,19,0,0,5,59,0,0,0,0,14,19,0,0,5,59] >;

C4.D36 in GAP, Magma, Sage, TeX

C_4.D_{36}
% in TeX

G:=Group("C4.D36");
// GroupNames label

G:=SmallGroup(288,30);
// by ID

G=gap.SmallGroup(288,30);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^4=c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^3>;
// generators/relations

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