C2^3.35D20 - GroupNames

G = C₂³.₃₅D₂₀ order 320 = 2⁶·5

6^th non-split extension by C₂³ of D₂₀ acting via D₂₀/D₁₀=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₃₅D₂₀, C₂².₃Dic₂₀, C₄₀⋊₅C₄⋊₂C₂, (C₂×C₈).₁D₁₀, (C₂×C₄).₃₀D₂₀, (C₂×C₂₀).₄₁D₄, C₂²⋊C₈.₃D₅, (C₂×C₁₀).₄Q₁₆, C₁₀.₅(C₂×Q₁₆), (C₂×C₄₀).₁C₂², C₂.₇(C₂×Dic₂₀), C₂₀.₄₄D₄⋊₄C₂, C₁₀.₇(C₈⋊C₂²), (C₂²×C₁₀).₄₉D₄, (C₂²×C₄).₇₆D₁₀, C₂₀.₂₈₀(C₄○D₄), C₂.₁₀(C₈⋊D₁₀), (C₂×C₂₀).₇₃₉C₂³, C₂₀.₄₈D₄.₃C₂, C₂².₁₀₂(C₂×D₂₀), C₅⋊₁(C₂³.₄₈D₄), C₄⋊Dic₅.₁₁C₂², C₄.₁₀₄(D₄⋊₂D₅), (C₂²×C₂₀).₄₉C₂², (C₂×Dic₁₀).₁₃C₂², C₁₀.₁₅(C₂².D₄), C₂.₁₁(C₂².D₂₀), (C₅×C₂²⋊C₈).₅C₂, (C₂×C₁₀).₁₂₂(C₂×D₄), (C₂×C₄⋊Dic₅).₁₃C₂, (C₂×C₄).₆₈₄(C₂²×D₅), SmallGroup(320,349)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₂³.₃₅D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄⋊Dic₅ — C₂×C₄⋊Dic₅ — C₂³.₃₅D₂₀

Lower central

C₅ — C₁₀ — C₂×C₂₀ — C₂³.₃₅D₂₀

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²⋊C₈

Generators and relations for C₂³.₃₅D₂₀
G = < a,b,c,d,e | a²=b²=c²=1, d²⁰=e²=c, ab=ba, ac=ca, dad^-1=eae^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=bd¹⁹ >

Subgroups: 398 in 104 conjugacy classes, 43 normal (25 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄₀, Dic₁₀, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂³.₄₈D₄, C₁₀.D₄, C₄⋊Dic₅, C₄⋊Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂×C₄₀, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂₀.₄₄D₄, C₄₀⋊₅C₄, C₅×C₂²⋊C₈, C₂₀.₄₈D₄, C₂×C₄⋊Dic₅, C₂³.₃₅D₂₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, Q₁₆, C₂×D₄, C₄○D₄, D₁₀, C₂².D₄, C₂×Q₁₆, C₈⋊C₂², D₂₀, C₂²×D₅, C₂³.₄₈D₄, Dic₂₀, C₂×D₂₀, D₄⋊₂D₅, C₂².D₂₀, C₂×Dic₂₀, C₈⋊D₁₀, C₂³.₃₅D₂₀

Smallest permutation representation of C₂³.₃₅D₂₀
►On 160 points

Generators in S₁₆₀

(2 149)(4 151)(6 153)(8 155)(10 157)(12 159)(14 121)(16 123)(18 125)(20 127)(22 129)(24 131)(26 133)(28 135)(30 137)(32 139)(34 141)(36 143)(38 145)(40 147)(41 91)(43 93)(45 95)(47 97)(49 99)(51 101)(53 103)(55 105)(57 107)(59 109)(61 111)(63 113)(65 115)(67 117)(69 119)(71 81)(73 83)(75 85)(77 87)(79 89)

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

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

(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)

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

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

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

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

59 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	5A	5B	8A	8B	8C	8D	10A	···	10F	10G	10H	10I	10J	20A	···	20H	20I	20J	20K	20L	40A	···	40P
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	4	5	5	8	8	8	8	10	···	10	10	10	10	10	20	···	20	20	20	20	20	40	···	40
size	1	1	1	1	2	2	2	2	4	20	20	20	20	40	40	2	2	4	4	4	4	2	···	2	4	4	4	4	2	···	2	4	4	4	4	4	···	4

59 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+		-	+	+	+	+	-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₅	C₄○D₄	Q₁₆	D₁₀	D₁₀	D₂₀	D₂₀	Dic₂₀	C₈⋊C₂²	D₄⋊₂D₅	C₈⋊D₁₀
kernel	C₂³.₃₅D₂₀	C₂₀.₄₄D₄	C₄₀⋊₅C₄	C₅×C₂²⋊C₈	C₂₀.₄₈D₄	C₂×C₄⋊Dic₅	C₂×C₂₀	C₂²×C₁₀	C₂²⋊C₈	C₂₀	C₂×C₁₀	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₁₀	C₄	C₂
# reps	1	2	2	1	1	1	1	1	2	4	4	4	2	4	4	16	1	4	4

Matrix representation of C₂³.₃₅D₂₀ ►in GL₆(𝔽₄₁)

1	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
0	0	0	0	40	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

40	0	0	0	0	0
0	40	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

12	29	0	0	0	0
12	12	0	0	0	0
0	0	27	30	0	0
0	0	11	32	0	0
0	0	0	0	40	39
0	0	0	0	1	1

21	3	0	0	0	0
3	20	0	0	0	0
0	0	7	34	0	0
0	0	1	34	0	0
0	0	0	0	9	18
0	0	0	0	32	32

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,29,12,0,0,0,0,0,0,27,11,0,0,0,0,30,32,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[21,3,0,0,0,0,3,20,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;

C₂³.₃₅D₂₀ in GAP, Magma, Sage, TeX

C_2^3._{35}D_{20}

% in TeX

G:=Group("C2^3.35D20");

// GroupNames label

G:=SmallGroup(320,349);

// by ID

G=gap.SmallGroup(320,349);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,254,219,310,1123,136,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;

// generators/relations

G = C23.35D20 order 320 = 26·5

6th non-split extension by C23 of D20 acting via D20/D10=C2

G = C₂³.₃₅D₂₀ order 320 = 2⁶·5

6^th non-split extension by C₂³ of D₂₀ acting via D₂₀/D₁₀=C₂