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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 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C9, Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D9 [×2], C18 [×3], C18 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, Dic9 [×2], C36 [×2], C36, D18 [×6], C2×C18, C2×C18 [×2], C2×C18 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, D36 [×2], C2×Dic9 [×2], C9⋊D4 [×4], C2×C36 [×2], C2×C36 [×2], C22×D9 [×2], C22×C18, C127D4, C4⋊Dic9, D18⋊C4 [×2], C2×D36, C2×C9⋊D4 [×2], C22×C36, C367D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D9, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, D18 [×3], C2×D12, C4○D12, C2×C3⋊D4, D36 [×2], C9⋊D4 [×2], 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 99 68 140)(2 98 69 139)(3 97 70 138)(4 96 71 137)(5 95 72 136)(6 94 37 135)(7 93 38 134)(8 92 39 133)(9 91 40 132)(10 90 41 131)(11 89 42 130)(12 88 43 129)(13 87 44 128)(14 86 45 127)(15 85 46 126)(16 84 47 125)(17 83 48 124)(18 82 49 123)(19 81 50 122)(20 80 51 121)(21 79 52 120)(22 78 53 119)(23 77 54 118)(24 76 55 117)(25 75 56 116)(26 74 57 115)(27 73 58 114)(28 108 59 113)(29 107 60 112)(30 106 61 111)(31 105 62 110)(32 104 63 109)(33 103 64 144)(34 102 65 143)(35 101 66 142)(36 100 67 141)
(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 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(64 72)(65 71)(66 70)(67 69)(73 130)(74 129)(75 128)(76 127)(77 126)(78 125)(79 124)(80 123)(81 122)(82 121)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)

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

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

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

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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