SL(2,3).F5 - GroupNames

G = SL₂(𝔽₃).F₅ order 480 = 2⁵·3·5

The non-split extension by SL₂(𝔽₃) of F₅ acting through Inn(SL₂(𝔽₃))

Aliases: SL₂(𝔽₃).F₅, C₅⋊C₈.A₄, C₅⋊(C₈.A₄), Q₈.F₅⋊C₃, Q₈.(C₃×F₅), (C₅×Q₈).C₁₂, C₂.₂(A₄×F₅), C₁₀.₁(C₄×A₄), Q₈⋊₂D₅.₂C₆, Dic₅.₄(C₂×A₄), (C₅×SL₂(𝔽₃)).C₄, Dic₅.A₄.₃C₂, SmallGroup(480,964)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₅×Q₈ — SL₂(𝔽₃).F₅

Chief series

C₁ — C₂ — C₁₀ — C₅×Q₈ — Q₈⋊₂D₅ — Dic₅.A₄ — SL₂(𝔽₃).F₅

Lower central

C₅×Q₈ — SL₂(𝔽₃).F₅

Upper central

C₁ — C₂

Generators and relations for SL₂(𝔽₃).F₅
G = < a,b,c,d,e | a⁴=c³=d⁵=1, b²=e⁴=a², bab^-1=a^-1, cac^-1=b, ad=da, ae=ea, cbc^-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

C₁

C₂

₃₀C₂

₄C₃

C₅

₃C₄

₅C₄

₁₅C₂²

₄C₆

C₁₀

₆D₅

₄C₁₅

Q₈

₅C₈

₁₅C₈

₁₅C₂×C₄

₁₅D₄

₂₀C₁₂

Dic₅

₃D₁₀

₃C₂₀

₄C₃₀

₅C₄○D₄

₁₅C₂×C₈

₁₅M₄(2)

SL₂(𝔽₃)

₂₀C₂₄

C₅⋊C₈

C₅×Q₈

₃D₂₀

₃C₅⋊C₈

₃C₄×D₅

₄C₃×Dic₅

₅C₈○D₄

₅C₄.A₄

Q₈⋊₂D₅

₃D₅⋊C₈

₃C₄.F₅

C₅×SL₂(𝔽₃)

₄C₃×C₅⋊C₈

₅C₈.A₄

Q₈.F₅

Dic₅.A₄

SL₂(𝔽₃).F₅

Smallest permutation representation of SL₂(𝔽₃).F₅
►On 160 points

Generators in S₁₆₀

(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 147 13 151)(10 148 14 152)(11 149 15 145)(12 150 16 146)(17 65 21 69)(18 66 22 70)(19 67 23 71)(20 68 24 72)(25 127 29 123)(26 128 30 124)(27 121 31 125)(28 122 32 126)(41 143 45 139)(42 144 46 140)(43 137 47 141)(44 138 48 142)(49 107 53 111)(50 108 54 112)(51 109 55 105)(52 110 56 106)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(73 131 77 135)(74 132 78 136)(75 133 79 129)(76 134 80 130)(89 117 93 113)(90 118 94 114)(91 119 95 115)(92 120 96 116)(97 155 101 159)(98 156 102 160)(99 157 103 153)(100 158 104 154)

(1 139 5 143)(2 140 6 144)(3 141 7 137)(4 142 8 138)(9 81 13 85)(10 82 14 86)(11 83 15 87)(12 84 16 88)(17 103 21 99)(18 104 22 100)(19 97 23 101)(20 98 24 102)(25 89 29 93)(26 90 30 94)(27 91 31 95)(28 92 32 96)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)(49 73 53 77)(50 74 54 78)(51 75 55 79)(52 76 56 80)(57 149 61 145)(58 150 62 146)(59 151 63 147)(60 152 64 148)(65 157 69 153)(66 158 70 154)(67 159 71 155)(68 160 72 156)(105 133 109 129)(106 134 110 130)(107 135 111 131)(108 136 112 132)(113 123 117 127)(114 124 118 128)(115 125 119 121)(116 126 120 122)

(9 147 85)(10 148 86)(11 149 87)(12 150 88)(13 151 81)(14 152 82)(15 145 83)(16 146 84)(17 103 153)(18 104 154)(19 97 155)(20 98 156)(21 99 157)(22 100 158)(23 101 159)(24 102 160)(25 113 123)(26 114 124)(27 115 125)(28 116 126)(29 117 127)(30 118 128)(31 119 121)(32 120 122)(33 43 137)(34 44 138)(35 45 139)(36 46 140)(37 47 141)(38 48 142)(39 41 143)(40 42 144)(49 135 111)(50 136 112)(51 129 105)(52 130 106)(53 131 107)(54 132 108)(55 133 109)(56 134 110)

(1 63 71 79 95)(2 80 64 96 72)(3 89 73 65 57)(4 66 90 58 74)(5 59 67 75 91)(6 76 60 92 68)(7 93 77 69 61)(8 70 94 62 78)(9 97 105 125 45)(10 126 98 46 106)(11 47 127 107 99)(12 108 48 100 128)(13 101 109 121 41)(14 122 102 42 110)(15 43 123 111 103)(16 112 44 104 124)(17 83 33 113 135)(18 114 84 136 34)(19 129 115 35 85)(20 36 130 86 116)(21 87 37 117 131)(22 118 88 132 38)(23 133 119 39 81)(24 40 134 82 120)(25 49 153 145 137)(26 146 50 138 154)(27 139 147 155 51)(28 156 140 52 148)(29 53 157 149 141)(30 150 54 142 158)(31 143 151 159 55)(32 160 144 56 152)

(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)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)

G:=sub<Sym(160)| (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,147,13,151)(10,148,14,152)(11,149,15,145)(12,150,16,146)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,127,29,123)(26,128,30,124)(27,121,31,125)(28,122,32,126)(41,143,45,139)(42,144,46,140)(43,137,47,141)(44,138,48,142)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,131,77,135)(74,132,78,136)(75,133,79,129)(76,134,80,130)(89,117,93,113)(90,118,94,114)(91,119,95,115)(92,120,96,116)(97,155,101,159)(98,156,102,160)(99,157,103,153)(100,158,104,154), (1,139,5,143)(2,140,6,144)(3,141,7,137)(4,142,8,138)(9,81,13,85)(10,82,14,86)(11,83,15,87)(12,84,16,88)(17,103,21,99)(18,104,22,100)(19,97,23,101)(20,98,24,102)(25,89,29,93)(26,90,30,94)(27,91,31,95)(28,92,32,96)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80)(57,149,61,145)(58,150,62,146)(59,151,63,147)(60,152,64,148)(65,157,69,153)(66,158,70,154)(67,159,71,155)(68,160,72,156)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(113,123,117,127)(114,124,118,128)(115,125,119,121)(116,126,120,122), (9,147,85)(10,148,86)(11,149,87)(12,150,88)(13,151,81)(14,152,82)(15,145,83)(16,146,84)(17,103,153)(18,104,154)(19,97,155)(20,98,156)(21,99,157)(22,100,158)(23,101,159)(24,102,160)(25,113,123)(26,114,124)(27,115,125)(28,116,126)(29,117,127)(30,118,128)(31,119,121)(32,120,122)(33,43,137)(34,44,138)(35,45,139)(36,46,140)(37,47,141)(38,48,142)(39,41,143)(40,42,144)(49,135,111)(50,136,112)(51,129,105)(52,130,106)(53,131,107)(54,132,108)(55,133,109)(56,134,110), (1,63,71,79,95)(2,80,64,96,72)(3,89,73,65,57)(4,66,90,58,74)(5,59,67,75,91)(6,76,60,92,68)(7,93,77,69,61)(8,70,94,62,78)(9,97,105,125,45)(10,126,98,46,106)(11,47,127,107,99)(12,108,48,100,128)(13,101,109,121,41)(14,122,102,42,110)(15,43,123,111,103)(16,112,44,104,124)(17,83,33,113,135)(18,114,84,136,34)(19,129,115,35,85)(20,36,130,86,116)(21,87,37,117,131)(22,118,88,132,38)(23,133,119,39,81)(24,40,134,82,120)(25,49,153,145,137)(26,146,50,138,154)(27,139,147,155,51)(28,156,140,52,148)(29,53,157,149,141)(30,150,54,142,158)(31,143,151,159,55)(32,160,144,56,152), (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)>;

G:=Group( (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,147,13,151)(10,148,14,152)(11,149,15,145)(12,150,16,146)(17,65,21,69)(18,66,22,70)(19,67,23,71)(20,68,24,72)(25,127,29,123)(26,128,30,124)(27,121,31,125)(28,122,32,126)(41,143,45,139)(42,144,46,140)(43,137,47,141)(44,138,48,142)(49,107,53,111)(50,108,54,112)(51,109,55,105)(52,110,56,106)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,131,77,135)(74,132,78,136)(75,133,79,129)(76,134,80,130)(89,117,93,113)(90,118,94,114)(91,119,95,115)(92,120,96,116)(97,155,101,159)(98,156,102,160)(99,157,103,153)(100,158,104,154), (1,139,5,143)(2,140,6,144)(3,141,7,137)(4,142,8,138)(9,81,13,85)(10,82,14,86)(11,83,15,87)(12,84,16,88)(17,103,21,99)(18,104,22,100)(19,97,23,101)(20,98,24,102)(25,89,29,93)(26,90,30,94)(27,91,31,95)(28,92,32,96)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,73,53,77)(50,74,54,78)(51,75,55,79)(52,76,56,80)(57,149,61,145)(58,150,62,146)(59,151,63,147)(60,152,64,148)(65,157,69,153)(66,158,70,154)(67,159,71,155)(68,160,72,156)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(113,123,117,127)(114,124,118,128)(115,125,119,121)(116,126,120,122), (9,147,85)(10,148,86)(11,149,87)(12,150,88)(13,151,81)(14,152,82)(15,145,83)(16,146,84)(17,103,153)(18,104,154)(19,97,155)(20,98,156)(21,99,157)(22,100,158)(23,101,159)(24,102,160)(25,113,123)(26,114,124)(27,115,125)(28,116,126)(29,117,127)(30,118,128)(31,119,121)(32,120,122)(33,43,137)(34,44,138)(35,45,139)(36,46,140)(37,47,141)(38,48,142)(39,41,143)(40,42,144)(49,135,111)(50,136,112)(51,129,105)(52,130,106)(53,131,107)(54,132,108)(55,133,109)(56,134,110), (1,63,71,79,95)(2,80,64,96,72)(3,89,73,65,57)(4,66,90,58,74)(5,59,67,75,91)(6,76,60,92,68)(7,93,77,69,61)(8,70,94,62,78)(9,97,105,125,45)(10,126,98,46,106)(11,47,127,107,99)(12,108,48,100,128)(13,101,109,121,41)(14,122,102,42,110)(15,43,123,111,103)(16,112,44,104,124)(17,83,33,113,135)(18,114,84,136,34)(19,129,115,35,85)(20,36,130,86,116)(21,87,37,117,131)(22,118,88,132,38)(23,133,119,39,81)(24,40,134,82,120)(25,49,153,145,137)(26,146,50,138,154)(27,139,147,155,51)(28,156,140,52,148)(29,53,157,149,141)(30,150,54,142,158)(31,143,151,159,55)(32,160,144,56,152), (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160) );

G=PermutationGroup([[(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,147,13,151),(10,148,14,152),(11,149,15,145),(12,150,16,146),(17,65,21,69),(18,66,22,70),(19,67,23,71),(20,68,24,72),(25,127,29,123),(26,128,30,124),(27,121,31,125),(28,122,32,126),(41,143,45,139),(42,144,46,140),(43,137,47,141),(44,138,48,142),(49,107,53,111),(50,108,54,112),(51,109,55,105),(52,110,56,106),(57,87,61,83),(58,88,62,84),(59,81,63,85),(60,82,64,86),(73,131,77,135),(74,132,78,136),(75,133,79,129),(76,134,80,130),(89,117,93,113),(90,118,94,114),(91,119,95,115),(92,120,96,116),(97,155,101,159),(98,156,102,160),(99,157,103,153),(100,158,104,154)], [(1,139,5,143),(2,140,6,144),(3,141,7,137),(4,142,8,138),(9,81,13,85),(10,82,14,86),(11,83,15,87),(12,84,16,88),(17,103,21,99),(18,104,22,100),(19,97,23,101),(20,98,24,102),(25,89,29,93),(26,90,30,94),(27,91,31,95),(28,92,32,96),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42),(49,73,53,77),(50,74,54,78),(51,75,55,79),(52,76,56,80),(57,149,61,145),(58,150,62,146),(59,151,63,147),(60,152,64,148),(65,157,69,153),(66,158,70,154),(67,159,71,155),(68,160,72,156),(105,133,109,129),(106,134,110,130),(107,135,111,131),(108,136,112,132),(113,123,117,127),(114,124,118,128),(115,125,119,121),(116,126,120,122)], [(9,147,85),(10,148,86),(11,149,87),(12,150,88),(13,151,81),(14,152,82),(15,145,83),(16,146,84),(17,103,153),(18,104,154),(19,97,155),(20,98,156),(21,99,157),(22,100,158),(23,101,159),(24,102,160),(25,113,123),(26,114,124),(27,115,125),(28,116,126),(29,117,127),(30,118,128),(31,119,121),(32,120,122),(33,43,137),(34,44,138),(35,45,139),(36,46,140),(37,47,141),(38,48,142),(39,41,143),(40,42,144),(49,135,111),(50,136,112),(51,129,105),(52,130,106),(53,131,107),(54,132,108),(55,133,109),(56,134,110)], [(1,63,71,79,95),(2,80,64,96,72),(3,89,73,65,57),(4,66,90,58,74),(5,59,67,75,91),(6,76,60,92,68),(7,93,77,69,61),(8,70,94,62,78),(9,97,105,125,45),(10,126,98,46,106),(11,47,127,107,99),(12,108,48,100,128),(13,101,109,121,41),(14,122,102,42,110),(15,43,123,111,103),(16,112,44,104,124),(17,83,33,113,135),(18,114,84,136,34),(19,129,115,35,85),(20,36,130,86,116),(21,87,37,117,131),(22,118,88,132,38),(23,133,119,39,81),(24,40,134,82,120),(25,49,153,145,137),(26,146,50,138,154),(27,139,147,155,51),(28,156,140,52,148),(29,53,157,149,141),(30,150,54,142,158),(31,143,151,159,55),(32,160,144,56,152)], [(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),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)]])

35 conjugacy classes

class	1	2A	2B	3A	3B	4A	4B	4C	5	6A	6B	8A	8B	8C	8D	8E	8F	10	12A	12B	12C	12D	15A	15B	20	24A	···	24H	30A	30B
order	1	2	2	3	3	4	4	4	5	6	6	8	8	8	8	8	8	10	12	12	12	12	15	15	20	24	···	24	30	30
size	1	1	30	4	4	5	5	6	4	4	4	5	5	5	5	30	30	4	20	20	20	20	16	16	24	20	···	20	16	16

35 irreducible representations

dim	1	1	1	1	1	1	12	2	3	3	3	4	4	8	8
type	+	+					+		+	+		+		+
image	C₁	C₂	C₃	C₄	C₆	C₁₂	A₄×F₅	C₈.A₄	A₄	C₂×A₄	C₄×A₄	F₅	C₃×F₅	SL₂(𝔽₃).F₅	SL₂(𝔽₃).F₅
kernel	SL₂(𝔽₃).F₅	Dic₅.A₄	Q₈.F₅	C₅×SL₂(𝔽₃)	Q₈⋊₂D₅	C₅×Q₈	C₂	C₅	C₅⋊C₈	Dic₅	C₁₀	SL₂(𝔽₃)	Q₈	C₁	C₁
# reps	1	1	2	2	2	4	1	12	1	1	2	1	2	1	2

Matrix representation of SL₂(𝔽₃).F₅ ►in GL₆(𝔽₂₄₁)

16	15	0	0	0	0
15	225	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
240	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
15	225	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	240
0	0	1	0	0	240
0	0	0	1	0	240
0	0	0	0	1	240

30	0	0	0	0	0
0	30	0	0	0	0
0	0	83	115	232	211
0	0	74	85	73	53
0	0	156	168	188	44
0	0	30	159	158	126

G:=sub<GL(6,GF(241))| [16,15,0,0,0,0,15,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,15,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,83,74,156,30,0,0,115,85,168,159,0,0,232,73,188,158,0,0,211,53,44,126] >;

SL₂(𝔽₃).F₅ in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3).F_5

% in TeX

G:=Group("SL(2,3).F5");

// GroupNames label

G:=SmallGroup(480,964);

// by ID

G=gap.SmallGroup(480,964);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-5,-2,42,856,514,584,221,795,382,4037,1363]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^5=1,b^2=e^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;

// generators/relations

Export

Subgroup lattice of SL₂(𝔽₃).F₅ in TeX

30	0	0	0	0	0
0	30	0	0	0	0
0	0	83	115	232	211
0	0	74	85	73	53
0	0	156	168	188	44
0	0	30	159	158	126

30	0	0	0	0	0
0	30	0	0	0	0
0	0	83	115	232	211
0	0	74	85	73	53
0	0	156	168	188	44
0	0	30	159	158	126

G = SL2(𝔽3).F5 order 480 = 25·3·5

The non-split extension by SL2(𝔽3) of F5 acting through Inn(SL2(𝔽3))

G = SL₂(𝔽₃).F₅ order 480 = 2⁵·3·5

The non-split extension by SL₂(𝔽₃) of F₅ acting through Inn(SL₂(𝔽₃))

30	0	0	0	0	0
0	30	0	0	0	0
0	0	83	115	232	211
0	0	74	85	73	53
0	0	156	168	188	44
0	0	30	159	158	126