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G = Dic94D4order 288 = 25·32

1st semidirect product of Dic9 and D4 acting through Inn(Dic9)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic94D4, C23.19D18, C9⋊D4⋊C4, C92(C4×D4), C2.2(D4×D9), D18⋊C49C2, D182(C2×C4), C22⋊C47D9, C6.78(S3×D4), C222(C4×D9), Dic9⋊C49C2, Dic91(C2×C4), C18.18(C2×D4), (C2×C4).25D18, (C4×Dic9)⋊11C2, (C2×C12).177D6, C18.7(C22×C4), (C22×C6).41D6, C18.22(C4○D4), C2.2(D42D9), C3.(Dic34D4), (C2×C18).22C23, (C2×C36).54C22, (C22×Dic9)⋊1C2, C6.75(D42S3), C22.14(C22×D9), (C22×C18).11C22, (C2×Dic9).26C22, (C22×D9).16C22, (C2×C4×D9)⋊9C2, C2.9(C2×C4×D9), C6.46(S3×C2×C4), (C2×C18)⋊2(C2×C4), (C2×C6).7(C4×S3), (C9×C22⋊C4)⋊9C2, (C2×C9⋊D4).2C2, (C3×C22⋊C4).13S3, (C2×C6).179(C22×S3), SmallGroup(288,91)

Series: Derived Chief Lower central Upper central

C1C18 — Dic94D4
C1C3C9C18C2×C18C22×D9C2×C9⋊D4 — Dic94D4
C9C18 — Dic94D4
C1C22C22⋊C4

Generators and relations for Dic94D4
 G = < a,b,c,d | a18=c4=d2=1, b2=a9, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 141 conjugacy classes, 54 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C9, Dic3 [×5], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, D9 [×2], C18 [×3], C18 [×2], C4×S3 [×2], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C4×D4, Dic9 [×4], Dic9, C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×2], C2×C18 [×2], C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×D9 [×2], C2×Dic9 [×3], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36 [×2], C22×D9, C22×C18, Dic34D4, C4×Dic9, Dic9⋊C4, D18⋊C4, C9×C22⋊C4, C2×C4×D9, C22×Dic9, C2×C9⋊D4, Dic94D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, D9, C4×S3 [×2], C22×S3, C4×D4, D18 [×3], S3×C2×C4, S3×D4, D42S3, C4×D9 [×2], C22×D9, Dic34D4, C2×C4×D9, D4×D9, D42D9, Dic94D4

Smallest permutation representation of Dic94D4
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 69 10 60)(2 68 11 59)(3 67 12 58)(4 66 13 57)(5 65 14 56)(6 64 15 55)(7 63 16 72)(8 62 17 71)(9 61 18 70)(19 144 28 135)(20 143 29 134)(21 142 30 133)(22 141 31 132)(23 140 32 131)(24 139 33 130)(25 138 34 129)(26 137 35 128)(27 136 36 127)(37 117 46 126)(38 116 47 125)(39 115 48 124)(40 114 49 123)(41 113 50 122)(42 112 51 121)(43 111 52 120)(44 110 53 119)(45 109 54 118)(73 98 82 107)(74 97 83 106)(75 96 84 105)(76 95 85 104)(77 94 86 103)(78 93 87 102)(79 92 88 101)(80 91 89 100)(81 108 90 99)
(1 49 35 81)(2 48 36 80)(3 47 19 79)(4 46 20 78)(5 45 21 77)(6 44 22 76)(7 43 23 75)(8 42 24 74)(9 41 25 73)(10 40 26 90)(11 39 27 89)(12 38 28 88)(13 37 29 87)(14 54 30 86)(15 53 31 85)(16 52 32 84)(17 51 33 83)(18 50 34 82)(55 119 132 104)(56 118 133 103)(57 117 134 102)(58 116 135 101)(59 115 136 100)(60 114 137 99)(61 113 138 98)(62 112 139 97)(63 111 140 96)(64 110 141 95)(65 109 142 94)(66 126 143 93)(67 125 144 92)(68 124 127 91)(69 123 128 108)(70 122 129 107)(71 121 130 106)(72 120 131 105)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 51)(20 52)(21 53)(22 54)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 91)(71 92)(72 93)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 127)(123 128)(124 129)(125 130)(126 131)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E9A9B9C12A12B12C12D18A···18I18J···18O36A···36L
order122222223444444444444666669991212121218···1818···1836···36
size1111221818222229999181818182224422244442···24···44···4

60 irreducible representations

dim11111111122222222224444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4S3D4D6D6C4○D4D9C4×S3D18D18C4×D9S3×D4D42S3D4×D9D42D9
kernelDic94D4C4×Dic9Dic9⋊C4D18⋊C4C9×C22⋊C4C2×C4×D9C22×Dic9C2×C9⋊D4C9⋊D4C3×C22⋊C4Dic9C2×C12C22×C6C18C22⋊C4C2×C6C2×C4C23C22C6C6C2C2
# reps111111118122123463121133

Matrix representation of Dic94D4 in GL4(𝔽37) generated by

113100
61700
0010
0001
,
9100
292800
00360
00036
,
203100
111700
00223
001115
,
1000
0100
00223
002415
G:=sub<GL(4,GF(37))| [11,6,0,0,31,17,0,0,0,0,1,0,0,0,0,1],[9,29,0,0,1,28,0,0,0,0,36,0,0,0,0,36],[20,11,0,0,31,17,0,0,0,0,22,11,0,0,3,15],[1,0,0,0,0,1,0,0,0,0,22,24,0,0,3,15] >;

Dic94D4 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes_4D_4
% in TeX

G:=Group("Dic9:4D4");
// GroupNames label

G:=SmallGroup(288,91);
// by ID

G=gap.SmallGroup(288,91);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,6725,292,9414]);
// Polycyclic

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

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