Copied to
clipboard

## G = C36.C23order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C36.C23
 Chief series C1 — C3 — C9 — C18 — C36 — D36 — D36⋊5C2 — C36.C23
 Lower central C9 — C18 — C36 — C36.C23
 Upper central C1 — C2 — C2×C4 — C2×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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, D9, C18, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C8.C22, Dic9, C36, C36, D18, C2×C18, C4.Dic3, Q82S3, C3⋊Q16, C4○D12, C6×Q8, C9⋊C8, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, Q8×C9, Q8×C9, Q8.11D6, C4.Dic9, C9⋊Q16, Q82D9, D365C2, Q8×C18, C36.C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C8.C22, D18, C2×C3⋊D4, C9⋊D4, 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 44)(38 43)(39 42)(40 41)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(106 108)(109 118)(110 117)(111 116)(112 115)(113 114)(119 144)(120 143)(121 142)(122 141)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)
(1 107 19 89)(2 108 20 90)(3 73 21 91)(4 74 22 92)(5 75 23 93)(6 76 24 94)(7 77 25 95)(8 78 26 96)(9 79 27 97)(10 80 28 98)(11 81 29 99)(12 82 30 100)(13 83 31 101)(14 84 32 102)(15 85 33 103)(16 86 34 104)(17 87 35 105)(18 88 36 106)(37 110 55 128)(38 111 56 129)(39 112 57 130)(40 113 58 131)(41 114 59 132)(42 115 60 133)(43 116 61 134)(44 117 62 135)(45 118 63 136)(46 119 64 137)(47 120 65 138)(48 121 66 139)(49 122 67 140)(50 123 68 141)(51 124 69 142)(52 125 70 143)(53 126 71 144)(54 127 72 109)
(1 72 19 54)(2 55 20 37)(3 38 21 56)(4 57 22 39)(5 40 23 58)(6 59 24 41)(7 42 25 60)(8 61 26 43)(9 44 27 62)(10 63 28 45)(11 46 29 64)(12 65 30 47)(13 48 31 66)(14 67 32 49)(15 50 33 68)(16 69 34 51)(17 52 35 70)(18 71 36 53)(73 129 91 111)(74 112 92 130)(75 131 93 113)(76 114 94 132)(77 133 95 115)(78 116 96 134)(79 135 97 117)(80 118 98 136)(81 137 99 119)(82 120 100 138)(83 139 101 121)(84 122 102 140)(85 141 103 123)(86 124 104 142)(87 143 105 125)(88 126 106 144)(89 109 107 127)(90 128 108 110)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 6 6 6 8 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 2 36 2 2 2 4 4 36 2 2 2 36 36 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D9 C3⋊D4 C3⋊D4 D18 D18 C9⋊D4 C9⋊D4 C8.C22 Q8.11D6 C36.C23 kernel C36.C23 C4.Dic9 C9⋊Q16 Q8⋊2D9 D36⋊5C2 Q8×C18 C6×Q8 C36 C2×C18 C2×C12 C3×Q8 C2×Q8 C12 C2×C6 C2×C4 Q8 C4 C22 C9 C3 C1 # reps 1 1 2 2 1 1 1 1 1 1 2 3 2 2 3 6 6 6 1 2 6

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

 28 70 47 41 3 31 58 47 48 34 42 3 14 48 70 45
,
 1 0 0 0 72 72 0 0 71 72 0 72 72 1 72 0
,
 30 60 13 60 13 43 0 13 56 56 30 13 0 56 60 43
,
 62 0 70 20 0 62 56 70 41 30 11 0 11 41 0 11
`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`

׿
×
𝔽