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G = C362D4order 288 = 25·32

2nd semidirect product of C36 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C362D4, D183D4, C23.13D18, (C2×D4)⋊4D9, (D4×C18)⋊3C2, C94(C4⋊D4), C42(C9⋊D4), (C6×D4).8S3, C2.26(D4×D9), C4⋊Dic914C2, C3.(D63D4), (C2×C4).52D18, C6.101(S3×D4), C18.51(C2×D4), (C2×C12).60D6, (C22×C6).51D6, C18.31(C4○D4), C12.13(C3⋊D4), (C2×C36).38C22, (C2×C18).53C23, C6.88(D42S3), C2.17(D42D9), C18.D411C2, C22.60(C22×D9), (C22×C18).20C22, (C2×Dic9).17C22, (C22×D9).26C22, (C2×C4×D9)⋊2C2, (C2×C9⋊D4)⋊5C2, C6.98(C2×C3⋊D4), C2.14(C2×C9⋊D4), (C2×C6).210(C22×S3), SmallGroup(288,148)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C362D4
C1C3C9C18C2×C18C22×D9C2×C4×D9 — C362D4
C9C2×C18 — C362D4
C1C22C2×D4

Generators and relations for C362D4
 G = < a,b,c | a36=b4=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >

Subgroups: 612 in 141 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, C9, Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×6], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], D9 [×2], C18 [×3], C18 [×2], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4⋊D4, Dic9 [×3], C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C18 [×6], C4⋊Dic3, C6.D4 [×2], S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, C4×D9 [×2], C2×Dic9, C2×Dic9 [×2], C9⋊D4 [×4], C2×C36, D4×C9 [×2], C22×D9, C22×C18 [×2], D63D4, C4⋊Dic9, C18.D4 [×2], C2×C4×D9, C2×C9⋊D4 [×2], D4×C18, C362D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, C3⋊D4 [×2], C22×S3, C4⋊D4, D18 [×3], S3×D4, D42S3, C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D63D4, D4×D9, D42D9, C2×C9⋊D4, C362D4

Smallest permutation representation of C362D4
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 90 123 63)(2 89 124 62)(3 88 125 61)(4 87 126 60)(5 86 127 59)(6 85 128 58)(7 84 129 57)(8 83 130 56)(9 82 131 55)(10 81 132 54)(11 80 133 53)(12 79 134 52)(13 78 135 51)(14 77 136 50)(15 76 137 49)(16 75 138 48)(17 74 139 47)(18 73 140 46)(19 108 141 45)(20 107 142 44)(21 106 143 43)(22 105 144 42)(23 104 109 41)(24 103 110 40)(25 102 111 39)(26 101 112 38)(27 100 113 37)(28 99 114 72)(29 98 115 71)(30 97 116 70)(31 96 117 69)(32 95 118 68)(33 94 119 67)(34 93 120 66)(35 92 121 65)(36 91 122 64)
(2 18)(3 35)(4 16)(5 33)(6 14)(7 31)(8 12)(9 29)(11 27)(13 25)(15 23)(17 21)(20 36)(22 34)(24 32)(26 30)(37 80)(38 97)(39 78)(40 95)(41 76)(42 93)(43 74)(44 91)(45 108)(46 89)(47 106)(48 87)(49 104)(50 85)(51 102)(52 83)(53 100)(54 81)(55 98)(56 79)(57 96)(58 77)(59 94)(60 75)(61 92)(62 73)(63 90)(64 107)(65 88)(66 105)(67 86)(68 103)(69 84)(70 101)(71 82)(72 99)(109 137)(110 118)(111 135)(112 116)(113 133)(115 131)(117 129)(119 127)(120 144)(121 125)(122 142)(124 140)(126 138)(128 136)(130 134)(139 143)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A6B6C6D6E6F6G9A9B9C12A12B18A···18I18J···18U36A···36F
order1222222234444446666666999121218···1818···1836···36
size1111441818222181836362224444222442···24···44···4

54 irreducible representations

dim111111222222222224444
type+++++++++++++++-+-
imageC1C2C2C2C2C2S3D4D4D6D6C4○D4D9C3⋊D4D18D18C9⋊D4S3×D4D42S3D4×D9D42D9
kernelC362D4C4⋊Dic9C18.D4C2×C4×D9C2×C9⋊D4D4×C18C6×D4C36D18C2×C12C22×C6C18C2×D4C12C2×C4C23C4C6C6C2C2
# reps1121211221223436121133

Matrix representation of C362D4 in GL6(𝔽37)

17110000
2660000
001000
000100
0000310
000006
,
3600000
110000
0003600
001000
0000036
000010
,
100000
36360000
001000
0003600
000010
0000036

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

C362D4 in GAP, Magma, Sage, TeX

C_{36}\rtimes_2D_4
% in TeX

G:=Group("C36:2D4");
// GroupNames label

G:=SmallGroup(288,148);
// by ID

G=gap.SmallGroup(288,148);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations

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