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G = C36.C23order 288 = 25·32

16th non-split extension by C36 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.20D4, Q8.12D18, C36.16C23, D36.10C22, Dic18.10C22, (C2×Q8)⋊4D9, (Q8×C18)⋊2C2, C9⋊Q165C2, C9⋊C8.3C22, Q82D95C2, (C2×C12).64D6, (C2×C4).17D18, C18.55(C2×D4), (C2×C18).43D4, C94(C8.C22), C4.Dic97C2, (C6×Q8).13S3, (C3×Q8).55D6, C4.17(C9⋊D4), C4.16(C22×D9), D365C2.5C2, (Q8×C9).7C22, C12.17(C3⋊D4), (C2×C36).42C22, C12.55(C22×S3), C3.(Q8.11D6), C22.11(C9⋊D4), C2.19(C2×C9⋊D4), C6.103(C2×C3⋊D4), (C2×C6).82(C3⋊D4), SmallGroup(288,153)

Series: Derived Chief Lower central Upper central

C1C36 — C36.C23
C1C3C9C18C36D36D365C2 — C36.C23
C9C18C36 — C36.C23
C1C2C2×C4C2×Q8

Generators and relations for C36.C23
 G = < a,b,c,d | a36=b2=1, c2=d2=a18, bab=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a9b, dcd-1=a18c >

Subgroups: 356 in 90 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], C9, Dic3, C12 [×2], C12 [×2], D6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, D9, C18, C18, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C8.C22, Dic9, C36 [×2], C36 [×2], D18, C2×C18, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, C9⋊C8 [×2], Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, Q8×C9 [×2], Q8×C9, Q8.11D6, C4.Dic9, C9⋊Q16 [×2], Q82D9 [×2], D365C2, Q8×C18, C36.C23
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8.C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Q8.11D6, C2×C9⋊D4, C36.C23

Smallest permutation representation of C36.C23
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)
(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 40)(38 39)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(109 136)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(137 144)(138 143)(139 142)(140 141)
(1 100 19 82)(2 101 20 83)(3 102 21 84)(4 103 22 85)(5 104 23 86)(6 105 24 87)(7 106 25 88)(8 107 26 89)(9 108 27 90)(10 73 28 91)(11 74 29 92)(12 75 30 93)(13 76 31 94)(14 77 32 95)(15 78 33 96)(16 79 34 97)(17 80 35 98)(18 81 36 99)(37 121 55 139)(38 122 56 140)(39 123 57 141)(40 124 58 142)(41 125 59 143)(42 126 60 144)(43 127 61 109)(44 128 62 110)(45 129 63 111)(46 130 64 112)(47 131 65 113)(48 132 66 114)(49 133 67 115)(50 134 68 116)(51 135 69 117)(52 136 70 118)(53 137 71 119)(54 138 72 120)
(1 52 19 70)(2 71 20 53)(3 54 21 72)(4 37 22 55)(5 56 23 38)(6 39 24 57)(7 58 25 40)(8 41 26 59)(9 60 27 42)(10 43 28 61)(11 62 29 44)(12 45 30 63)(13 64 31 46)(14 47 32 65)(15 66 33 48)(16 49 34 67)(17 68 35 50)(18 51 36 69)(73 109 91 127)(74 128 92 110)(75 111 93 129)(76 130 94 112)(77 113 95 131)(78 132 96 114)(79 115 97 133)(80 134 98 116)(81 117 99 135)(82 136 100 118)(83 119 101 137)(84 138 102 120)(85 121 103 139)(86 140 104 122)(87 123 105 141)(88 142 106 124)(89 125 107 143)(90 144 108 126)

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

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

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

51 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E6A6B6C8A8B9A9B9C12A···12F18A···18I36A···36R
order12223444446668899912···1218···1836···36
size11236222443622236362224···42···24···4

51 irreducible representations

dim111111222222222222444
type++++++++++++++-
imageC1C2C2C2C2C2S3D4D4D6D6D9C3⋊D4C3⋊D4D18D18C9⋊D4C9⋊D4C8.C22Q8.11D6C36.C23
kernelC36.C23C4.Dic9C9⋊Q16Q82D9D365C2Q8×C18C6×Q8C36C2×C18C2×C12C3×Q8C2×Q8C12C2×C6C2×C4Q8C4C22C9C3C1
# reps112211111123223666126

Matrix representation of C36.C23 in GL4(𝔽73) generated by

28704741
3315847
4834423
14487045
,
1000
727200
7172072
721720
,
30601360
1343013
56563013
0566043
,
6207020
0625670
4130110
1141011
G:=sub<GL(4,GF(73))| [28,3,48,14,70,31,34,48,47,58,42,70,41,47,3,45],[1,72,71,72,0,72,72,1,0,0,0,72,0,0,72,0],[30,13,56,0,60,43,56,56,13,0,30,60,60,13,13,43],[62,0,41,11,0,62,30,41,70,56,11,0,20,70,0,11] >;

C36.C23 in GAP, Magma, Sage, TeX

C_{36}.C_2^3
% in TeX

G:=Group("C36.C2^3");
// GroupNames label

G:=SmallGroup(288,153);
// by ID

G=gap.SmallGroup(288,153);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,675,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^36=b^2=1,c^2=d^2=a^18,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=a^18*c>;
// generators/relations

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