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G = C367D4order 288 = 25·32

1st semidirect product of C36 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C367D4, C222D36, C23.29D18, (C2×C18)⋊5D4, D18⋊C43C2, (C2×D36)⋊6C2, C93(C4⋊D4), C43(C9⋊D4), C4⋊Dic99C2, (C22×C4)⋊6D9, (C22×C36)⋊6C2, C3.(C127D4), (C2×C6).31D12, (C2×C4).70D18, C2.17(C2×D36), C18.44(C2×D4), C6.46(C2×D12), (C2×C12).346D6, C6.89(C4○D12), C18.19(C4○D4), (C22×C12).20S3, (C2×C18).48C23, (C2×C36).78C22, (C22×C6).142D6, C12.111(C3⋊D4), C2.19(D365C2), C22.56(C22×D9), (C22×C18).40C22, (C2×Dic9).14C22, (C22×D9).10C22, (C2×C9⋊D4)⋊3C2, C2.7(C2×C9⋊D4), C6.91(C2×C3⋊D4), (C2×C6).205(C22×S3), SmallGroup(288,140)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C367D4
C1C3C9C18C2×C18C22×D9C2×D36 — C367D4
C9C2×C18 — C367D4
C1C22C22×C4

Generators and relations for C367D4
 G = < a,b,c | a36=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 692 in 141 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4⋊D4, Dic9, C36, C36, D18, C2×C18, C2×C18, C2×C18, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, D36, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C127D4, C4⋊Dic9, D18⋊C4, C2×D36, C2×C9⋊D4, C22×C36, C367D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, D12, C3⋊D4, C22×S3, C4⋊D4, D18, C2×D12, C4○D12, C2×C3⋊D4, D36, C9⋊D4, C22×D9, C127D4, C2×D36, D365C2, C2×C9⋊D4, C367D4

Smallest permutation representation of C367D4
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)
(1 109 92 59)(2 144 93 58)(3 143 94 57)(4 142 95 56)(5 141 96 55)(6 140 97 54)(7 139 98 53)(8 138 99 52)(9 137 100 51)(10 136 101 50)(11 135 102 49)(12 134 103 48)(13 133 104 47)(14 132 105 46)(15 131 106 45)(16 130 107 44)(17 129 108 43)(18 128 73 42)(19 127 74 41)(20 126 75 40)(21 125 76 39)(22 124 77 38)(23 123 78 37)(24 122 79 72)(25 121 80 71)(26 120 81 70)(27 119 82 69)(28 118 83 68)(29 117 84 67)(30 116 85 66)(31 115 86 65)(32 114 87 64)(33 113 88 63)(34 112 89 62)(35 111 90 61)(36 110 91 60)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 131)(38 130)(39 129)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 144)(61 143)(62 142)(63 141)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 133)(72 132)(73 75)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A···6G9A9B9C12A···12H18A···18U36A···36X
order1222222234444446···699912···1218···1836···36
size11112236362222236362···22222···22···22···2

78 irreducible representations

dim111111222222222222222
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6C4○D4D9C3⋊D4D12D18D18C4○D12C9⋊D4D36D365C2
kernelC367D4C4⋊Dic9D18⋊C4C2×D36C2×C9⋊D4C22×C36C22×C12C36C2×C18C2×C12C22×C6C18C22×C4C12C2×C6C2×C4C23C6C4C22C2
# reps112121122212344634121212

Matrix representation of C367D4 in GL4(𝔽37) generated by

03600
1100
002925
00124
,
302300
30700
00714
00730
,
1000
363600
00360
0011
G:=sub<GL(4,GF(37))| [0,1,0,0,36,1,0,0,0,0,29,12,0,0,25,4],[30,30,0,0,23,7,0,0,0,0,7,7,0,0,14,30],[1,36,0,0,0,36,0,0,0,0,36,1,0,0,0,1] >;

C367D4 in GAP, Magma, Sage, TeX

C_{36}\rtimes_7D_4
% in TeX

G:=Group("C36:7D4");
// GroupNames label

G:=SmallGroup(288,140);
// by ID

G=gap.SmallGroup(288,140);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,6725,292,9414]);
// Polycyclic

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

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