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G = D36⋊C4order 288 = 25·32

5th semidirect product of D36 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D365C4, Dic95D4, C93(C4×D4), C4⋊C48D9, C41(C4×D9), C362(C2×C4), C2.4(D4×D9), D183(C2×C4), C6.85(S3×D4), D18⋊C412C2, C12.10(C4×S3), (C4×Dic9)⋊3C2, (C2×D36).7C2, C18.24(C2×D4), (C2×C4).32D18, C3.(Dic35D4), (C2×C12).182D6, C18.33(C4○D4), (C2×C36).57C22, (C2×C18).34C23, C18.11(C22×C4), C2.2(Q83D9), C6.39(Q83S3), C22.18(C22×D9), (C2×Dic9).33C22, (C22×D9).19C22, (C9×C4⋊C4)⋊4C2, (C2×C4×D9)⋊12C2, C6.50(S3×C2×C4), C2.13(C2×C4×D9), (C3×C4⋊C4).11S3, (C2×C6).191(C22×S3), SmallGroup(288,103)

Series: Derived Chief Lower central Upper central

C1C18 — D36⋊C4
C1C3C9C18C2×C18C22×D9C2×D36 — D36⋊C4
C9C18 — D36⋊C4
C1C22C4⋊C4

Generators and relations for D36⋊C4
 G = < a,b,c | a36=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a18b >

Subgroups: 656 in 141 conjugacy classes, 54 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], C9, Dic3 [×3], C12 [×2], C12 [×2], 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, C2×C12 [×2], C22×S3 [×2], C4×D4, Dic9 [×2], Dic9, C36 [×2], C36 [×2], D18 [×4], D18 [×4], C2×C18, C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C4×D9 [×4], D36 [×4], C2×Dic9 [×2], C2×C36, C2×C36 [×2], C22×D9 [×2], Dic35D4, C4×Dic9, D18⋊C4 [×2], C9×C4⋊C4, C2×C4×D9 [×2], C2×D36, D36⋊C4
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], C22×S3, C4×D4, D18 [×3], S3×C2×C4, S3×D4, Q83S3, C4×D9 [×2], C22×D9, Dic35D4, C2×C4×D9, D4×D9, Q83D9, D36⋊C4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G4H4I4J4K4L6A6B6C9A9B9C12A···12F18A···18I36A···36R
order1222222234···444444466699912···1218···1836···36
size11111818181822···2999918182222224···42···24···4

60 irreducible representations

dim1111111222222224444
type+++++++++++++++
imageC1C2C2C2C2C2C4S3D4D6C4○D4D9C4×S3D18C4×D9S3×D4Q83S3D4×D9Q83D9
kernelD36⋊C4C4×Dic9D18⋊C4C9×C4⋊C4C2×C4×D9C2×D36D36C3×C4⋊C4Dic9C2×C12C18C4⋊C4C12C2×C4C4C6C6C2C2
# reps11212181232349121133

Matrix representation of D36⋊C4 in GL4(𝔽37) generated by

311100
262000
00362
00361
,
1100
03600
00360
00361
,
6000
0600
00135
00036
G:=sub<GL(4,GF(37))| [31,26,0,0,11,20,0,0,0,0,36,36,0,0,2,1],[1,0,0,0,1,36,0,0,0,0,36,36,0,0,0,1],[6,0,0,0,0,6,0,0,0,0,1,0,0,0,35,36] >;

D36⋊C4 in GAP, Magma, Sage, TeX

D_{36}\rtimes C_4
% in TeX

G:=Group("D36:C4");
// GroupNames label

G:=SmallGroup(288,103);
// by ID

G=gap.SmallGroup(288,103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,219,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations

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