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G = C4×D36order 288 = 25·32

Direct product of C4 and D36

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D36, C365D4, C425D9, C12.55D12, C91(C4×D4), C42(C4×D9), C3.(C4×D12), C364(C2×C4), (C4×C36)⋊7C2, D181(C2×C4), C2.1(C2×D36), C18.2(C2×D4), D18⋊C417C2, C12.58(C4×S3), (C4×C12).13S3, C4⋊Dic916C2, (C2×C4).97D18, C6.31(C2×D12), (C2×D36).10C2, C18.4(C4○D4), (C2×C12).368D6, C18.4(C22×C4), C6.74(C4○D12), (C2×C36).72C22, (C2×C18).14C23, C2.3(D365C2), C22.11(C22×D9), (C2×Dic9).22C22, (C22×D9).15C22, (C2×C4×D9)⋊7C2, C2.6(C2×C4×D9), C6.43(S3×C2×C4), (C2×C6).171(C22×S3), SmallGroup(288,83)

Series: Derived Chief Lower central Upper central

C1C18 — C4×D36
C1C3C9C18C2×C18C22×D9C2×D36 — C4×D36
C9C18 — C4×D36
C1C2×C4C42

Generators and relations for C4×D36
 G = < a,b,c | a4=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 656 in 141 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C9, Dic3 [×2], C12 [×4], C12, D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, D9 [×4], C18 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], C4×D4, Dic9 [×2], C36 [×4], C36, D18 [×4], D18 [×4], C2×C18, C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C4×D9 [×4], D36 [×4], C2×Dic9 [×2], C2×C36 [×3], C22×D9 [×2], C4×D12, C4⋊Dic9, D18⋊C4 [×2], C4×C36, C2×C4×D9 [×2], C2×D36, C4×D36
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], D12 [×2], C22×S3, C4×D4, D18 [×3], S3×C2×C4, C2×D12, C4○D12, C4×D9 [×2], D36 [×2], C22×D9, C4×D12, C2×C4×D9, C2×D36, D365C2, C4×D36

Smallest permutation representation of C4×D36
On 144 points
Generators in S144
(1 129 87 67)(2 130 88 68)(3 131 89 69)(4 132 90 70)(5 133 91 71)(6 134 92 72)(7 135 93 37)(8 136 94 38)(9 137 95 39)(10 138 96 40)(11 139 97 41)(12 140 98 42)(13 141 99 43)(14 142 100 44)(15 143 101 45)(16 144 102 46)(17 109 103 47)(18 110 104 48)(19 111 105 49)(20 112 106 50)(21 113 107 51)(22 114 108 52)(23 115 73 53)(24 116 74 54)(25 117 75 55)(26 118 76 56)(27 119 77 57)(28 120 78 58)(29 121 79 59)(30 122 80 60)(31 123 81 61)(32 124 82 62)(33 125 83 63)(34 126 84 64)(35 127 85 65)(36 128 86 66)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 113)(38 112)(39 111)(40 110)(41 109)(42 144)(43 143)(44 142)(45 141)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C9A9B9C12A···12L18A···18I36A···36AJ
order12222222344444444444466699912···1218···1836···36
size111118181818211112222181818182222222···22···22···2

84 irreducible representations

dim1111111222222222222
type+++++++++++++
imageC1C2C2C2C2C2C4S3D4D6C4○D4D9C4×S3D12D18C4○D12C4×D9D36D365C2
kernelC4×D36C4⋊Dic9D18⋊C4C4×C36C2×C4×D9C2×D36D36C4×C12C36C2×C12C18C42C12C12C2×C4C6C4C4C2
# reps1121218123234494121212

Matrix representation of C4×D36 in GL3(𝔽37) generated by

600
060
006
,
3600
048
02912
,
100
0620
02631
G:=sub<GL(3,GF(37))| [6,0,0,0,6,0,0,0,6],[36,0,0,0,4,29,0,8,12],[1,0,0,0,6,26,0,20,31] >;

C4×D36 in GAP, Magma, Sage, TeX

C_4\times D_{36}
% in TeX

G:=Group("C4xD36");
// GroupNames label

G:=SmallGroup(288,83);
// by ID

G=gap.SmallGroup(288,83);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,6725,292,9414]);
// Polycyclic

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

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