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G = C4⋊C47D9order 288 = 25·32

1st semidirect product of C4⋊C4 and D9 acting through Inn(C4⋊C4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C47D9, (C4×D9)⋊2C4, C12.9(C4×S3), C4.14(C4×D9), C36.11(C2×C4), D18⋊C4.3C2, C4⋊Dic912C2, D18.4(C2×C4), (C2×C4).31D18, C93(C42⋊C2), (C4×Dic9)⋊13C2, (C2×C12).181D6, Dic9.9(C2×C4), C18.26(C4○D4), C2.4(D42D9), C18.10(C22×C4), (C2×C18).33C23, (C2×C36).56C22, C2.1(Q83D9), C6.82(D42S3), C6.38(Q83S3), C22.17(C22×D9), (C2×Dic9).32C22, (C22×D9).18C22, (C9×C4⋊C4)⋊3C2, C6.49(S3×C2×C4), (C2×C4×D9).1C2, C2.12(C2×C4×D9), C3.(C4⋊C47S3), (C3×C4⋊C4).10S3, (C2×C6).190(C22×S3), SmallGroup(288,102)

Series: Derived Chief Lower central Upper central

C1C18 — C4⋊C47D9
C1C3C9C18C2×C18C22×D9C2×C4×D9 — C4⋊C47D9
C9C18 — C4⋊C47D9
C1C22C4⋊C4

Generators and relations for C4⋊C47D9
 G = < a,b,c,d | a4=b4=c9=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 448 in 114 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C9, Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, D9 [×2], C18 [×3], C4×S3 [×4], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C22×S3, C42⋊C2, Dic9 [×2], Dic9 [×2], C36 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C4×D9 [×4], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], C22×D9, C4⋊C47S3, C4×Dic9 [×2], C4⋊Dic9, D18⋊C4 [×2], C9×C4⋊C4, C2×C4×D9, C4⋊C47D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4○D4 [×2], D9, C4×S3 [×2], C22×S3, C42⋊C2, D18 [×3], S3×C2×C4, D42S3, Q83S3, C4×D9 [×2], C22×D9, C4⋊C47S3, C2×C4×D9, D42D9, Q83D9, C4⋊C47D9

Smallest permutation representation of C4⋊C47D9
On 144 points
Generators in S144
(1 104 14 95)(2 105 15 96)(3 106 16 97)(4 107 17 98)(5 108 18 99)(6 100 10 91)(7 101 11 92)(8 102 12 93)(9 103 13 94)(19 82 28 73)(20 83 29 74)(21 84 30 75)(22 85 31 76)(23 86 32 77)(24 87 33 78)(25 88 34 79)(26 89 35 80)(27 90 36 81)(37 136 46 127)(38 137 47 128)(39 138 48 129)(40 139 49 130)(41 140 50 131)(42 141 51 132)(43 142 52 133)(44 143 53 134)(45 144 54 135)(55 118 64 109)(56 119 65 110)(57 120 66 111)(58 121 67 112)(59 122 68 113)(60 123 69 114)(61 124 70 115)(62 125 71 116)(63 126 72 117)
(1 59 23 41)(2 60 24 42)(3 61 25 43)(4 62 26 44)(5 63 27 45)(6 55 19 37)(7 56 20 38)(8 57 21 39)(9 58 22 40)(10 64 28 46)(11 65 29 47)(12 66 30 48)(13 67 31 49)(14 68 32 50)(15 69 33 51)(16 70 34 52)(17 71 35 53)(18 72 36 54)(73 136 91 118)(74 137 92 119)(75 138 93 120)(76 139 94 121)(77 140 95 122)(78 141 96 123)(79 142 97 124)(80 143 98 125)(81 144 99 126)(82 127 100 109)(83 128 101 110)(84 129 102 111)(85 130 103 112)(86 131 104 113)(87 132 105 114)(88 133 106 115)(89 134 107 116)(90 135 108 117)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 80)(74 79)(75 78)(76 77)(82 89)(83 88)(84 87)(85 86)(91 98)(92 97)(93 96)(94 95)(100 107)(101 106)(102 105)(103 104)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(117 126)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(135 144)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H4I4J4K4L4M4N6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12222234···44444444466699912···1218···1836···36
size1111181822···29999181818182222224···42···24···4

60 irreducible representations

dim111111122222224444
type++++++++++-+-+
imageC1C2C2C2C2C2C4S3D6C4○D4D9C4×S3D18C4×D9D42S3Q83S3D42D9Q83D9
kernelC4⋊C47D9C4×Dic9C4⋊Dic9D18⋊C4C9×C4⋊C4C2×C4×D9C4×D9C3×C4⋊C4C2×C12C18C4⋊C4C12C2×C4C4C6C6C2C2
# reps1212118134349121133

Matrix representation of C4⋊C47D9 in GL5(𝔽37)

10000
036000
003600
000310
00006
,
60000
036000
003600
00001
000360
,
10000
0312000
0171100
00010
00001
,
360000
0312000
026600
00010
000036

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,31,0,0,0,0,0,6],[6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,1,0],[1,0,0,0,0,0,31,17,0,0,0,20,11,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,31,26,0,0,0,20,6,0,0,0,0,0,1,0,0,0,0,0,36] >;

C4⋊C47D9 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_7D_9
% in TeX

G:=Group("C4:C4:7D9");
// GroupNames label

G:=SmallGroup(288,102);
// by ID

G=gap.SmallGroup(288,102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,219,58,6725,292,9414]);
// Polycyclic

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

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