direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×Q8⋊5D4, C10.1182- (1+4), Q8⋊5(C5×D4), (C5×Q8)⋊23D4, (D4×C20)⋊46C2, (C4×D4)⋊17C10, (C4×Q8)⋊11C10, (Q8×C20)⋊31C2, C4.42(D4×C10), C4⋊D4⋊13C10, C20.403(C2×D4), C22⋊Q8⋊13C10, (C22×Q8)⋊6C10, C4.4D4⋊11C10, C42.43(C2×C10), (C2×C10).368C24, (C4×C20).284C22, (C2×C20).714C23, C10.196(C22×D4), (D4×C10).322C22, C22.42(C23×C10), C23.42(C22×C10), (Q8×C10).274C22, C2.10(C5×2- (1+4)), (C22×C20).454C22, (C22×C10).100C23, (Q8×C2×C10)⋊18C2, C2.20(D4×C2×C10), (C2×C4○D4)⋊8C10, C22⋊3(C5×C4○D4), (C10×C4○D4)⋊24C2, (C5×C4⋊D4)⋊40C2, C4⋊C4.71(C2×C10), C2.22(C10×C4○D4), (C5×C22⋊Q8)⋊40C2, (C2×C10)⋊17(C4○D4), (C2×D4).67(C2×C10), C10.241(C2×C4○D4), (C5×C4.4D4)⋊31C2, (C2×Q8).61(C2×C10), (C5×C4⋊C4).396C22, C22⋊C4.19(C2×C10), (C2×C4).60(C22×C10), (C22×C4).66(C2×C10), (C5×C22⋊C4).88C22, SmallGroup(320,1550)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 290 conjugacy classes, 166 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×9], C2×C4 [×12], D4 [×12], Q8 [×4], Q8 [×6], C23, C23 [×3], C10 [×3], C10 [×5], C42 [×3], C22⋊C4, C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×4], C20 [×6], C20 [×8], C2×C10, C2×C10 [×2], C2×C10 [×11], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C2×C20 [×2], C2×C20 [×9], C2×C20 [×12], C5×D4 [×12], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22×C10 [×3], Q8⋊5D4, C4×C20 [×3], C5×C22⋊C4, C5×C22⋊C4 [×9], C5×C4⋊C4 [×6], C22×C20 [×6], D4×C10 [×6], Q8×C10, Q8×C10 [×3], Q8×C10 [×4], C5×C4○D4 [×4], D4×C20 [×3], Q8×C20, C5×C4⋊D4 [×3], C5×C22⋊Q8 [×3], C5×C4.4D4 [×3], Q8×C2×C10, C10×C4○D4, C5×Q8⋊5D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2- (1+4), C5×D4 [×4], C22×C10 [×15], Q8⋊5D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2- (1+4), C5×Q8⋊5D4
Generators and relations
G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 160 points
Generators in S160
(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 47 35 41)(2 48 31 42)(3 49 32 43)(4 50 33 44)(5 46 34 45)(6 145 160 146)(7 141 156 147)(8 142 157 148)(9 143 158 149)(10 144 159 150)(11 151 17 137)(12 152 18 138)(13 153 19 139)(14 154 20 140)(15 155 16 136)(21 55 27 36)(22 51 28 37)(23 52 29 38)(24 53 30 39)(25 54 26 40)(56 87 75 81)(57 88 71 82)(58 89 72 83)(59 90 73 84)(60 86 74 85)(61 95 67 76)(62 91 68 77)(63 92 69 78)(64 93 70 79)(65 94 66 80)(96 121 115 127)(97 122 111 128)(98 123 112 129)(99 124 113 130)(100 125 114 126)(101 116 107 135)(102 117 108 131)(103 118 109 132)(104 119 110 133)(105 120 106 134)
(1 107 35 101)(2 108 31 102)(3 109 32 103)(4 110 33 104)(5 106 34 105)(6 86 160 85)(7 87 156 81)(8 88 157 82)(9 89 158 83)(10 90 159 84)(11 77 17 91)(12 78 18 92)(13 79 19 93)(14 80 20 94)(15 76 16 95)(21 115 27 96)(22 111 28 97)(23 112 29 98)(24 113 30 99)(25 114 26 100)(36 127 55 121)(37 128 51 122)(38 129 52 123)(39 130 53 124)(40 126 54 125)(41 135 47 116)(42 131 48 117)(43 132 49 118)(44 133 50 119)(45 134 46 120)(56 147 75 141)(57 148 71 142)(58 149 72 143)(59 150 73 144)(60 146 74 145)(61 155 67 136)(62 151 68 137)(63 152 69 138)(64 153 70 139)(65 154 66 140)
(1 136 21 141)(2 137 22 142)(3 138 23 143)(4 139 24 144)(5 140 25 145)(6 45 20 40)(7 41 16 36)(8 42 17 37)(9 43 18 38)(10 44 19 39)(11 51 157 48)(12 52 158 49)(13 53 159 50)(14 54 160 46)(15 55 156 47)(26 146 34 154)(27 147 35 155)(28 148 31 151)(29 149 32 152)(30 150 33 153)(56 101 61 96)(57 102 62 97)(58 103 63 98)(59 104 64 99)(60 105 65 100)(66 114 74 106)(67 115 75 107)(68 111 71 108)(69 112 72 109)(70 113 73 110)(76 127 81 135)(77 128 82 131)(78 129 83 132)(79 130 84 133)(80 126 85 134)(86 120 94 125)(87 116 95 121)(88 117 91 122)(89 118 92 123)(90 119 93 124)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 85)(7 81)(8 82)(9 83)(10 84)(11 91)(12 92)(13 93)(14 94)(15 95)(16 76)(17 77)(18 78)(19 79)(20 80)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 135)(37 131)(38 132)(39 133)(40 134)(41 127)(42 128)(43 129)(44 130)(45 126)(46 125)(47 121)(48 122)(49 123)(50 124)(51 117)(52 118)(53 119)(54 120)(55 116)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 154)(67 155)(68 151)(69 152)(70 153)(71 148)(72 149)(73 150)(74 146)(75 147)(86 160)(87 156)(88 157)(89 158)(90 159)
G:=sub<Sym(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), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,145,160,146)(7,141,156,147)(8,142,157,148)(9,143,158,149)(10,144,159,150)(11,151,17,137)(12,152,18,138)(13,153,19,139)(14,154,20,140)(15,155,16,136)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,121,115,127)(97,122,111,128)(98,123,112,129)(99,124,113,130)(100,125,114,126)(101,116,107,135)(102,117,108,131)(103,118,109,132)(104,119,110,133)(105,120,106,134), (1,107,35,101)(2,108,31,102)(3,109,32,103)(4,110,33,104)(5,106,34,105)(6,86,160,85)(7,87,156,81)(8,88,157,82)(9,89,158,83)(10,90,159,84)(11,77,17,91)(12,78,18,92)(13,79,19,93)(14,80,20,94)(15,76,16,95)(21,115,27,96)(22,111,28,97)(23,112,29,98)(24,113,30,99)(25,114,26,100)(36,127,55,121)(37,128,51,122)(38,129,52,123)(39,130,53,124)(40,126,54,125)(41,135,47,116)(42,131,48,117)(43,132,49,118)(44,133,50,119)(45,134,46,120)(56,147,75,141)(57,148,71,142)(58,149,72,143)(59,150,73,144)(60,146,74,145)(61,155,67,136)(62,151,68,137)(63,152,69,138)(64,153,70,139)(65,154,66,140), (1,136,21,141)(2,137,22,142)(3,138,23,143)(4,139,24,144)(5,140,25,145)(6,45,20,40)(7,41,16,36)(8,42,17,37)(9,43,18,38)(10,44,19,39)(11,51,157,48)(12,52,158,49)(13,53,159,50)(14,54,160,46)(15,55,156,47)(26,146,34,154)(27,147,35,155)(28,148,31,151)(29,149,32,152)(30,150,33,153)(56,101,61,96)(57,102,62,97)(58,103,63,98)(59,104,64,99)(60,105,65,100)(66,114,74,106)(67,115,75,107)(68,111,71,108)(69,112,72,109)(70,113,73,110)(76,127,81,135)(77,128,82,131)(78,129,83,132)(79,130,84,133)(80,126,85,134)(86,120,94,125)(87,116,95,121)(88,117,91,122)(89,118,92,123)(90,119,93,124), (1,96)(2,97)(3,98)(4,99)(5,100)(6,85)(7,81)(8,82)(9,83)(10,84)(11,91)(12,92)(13,93)(14,94)(15,95)(16,76)(17,77)(18,78)(19,79)(20,80)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,135)(37,131)(38,132)(39,133)(40,134)(41,127)(42,128)(43,129)(44,130)(45,126)(46,125)(47,121)(48,122)(49,123)(50,124)(51,117)(52,118)(53,119)(54,120)(55,116)(56,141)(57,142)(58,143)(59,144)(60,145)(61,136)(62,137)(63,138)(64,139)(65,140)(66,154)(67,155)(68,151)(69,152)(70,153)(71,148)(72,149)(73,150)(74,146)(75,147)(86,160)(87,156)(88,157)(89,158)(90,159)>;
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)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,145,160,146)(7,141,156,147)(8,142,157,148)(9,143,158,149)(10,144,159,150)(11,151,17,137)(12,152,18,138)(13,153,19,139)(14,154,20,140)(15,155,16,136)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,121,115,127)(97,122,111,128)(98,123,112,129)(99,124,113,130)(100,125,114,126)(101,116,107,135)(102,117,108,131)(103,118,109,132)(104,119,110,133)(105,120,106,134), (1,107,35,101)(2,108,31,102)(3,109,32,103)(4,110,33,104)(5,106,34,105)(6,86,160,85)(7,87,156,81)(8,88,157,82)(9,89,158,83)(10,90,159,84)(11,77,17,91)(12,78,18,92)(13,79,19,93)(14,80,20,94)(15,76,16,95)(21,115,27,96)(22,111,28,97)(23,112,29,98)(24,113,30,99)(25,114,26,100)(36,127,55,121)(37,128,51,122)(38,129,52,123)(39,130,53,124)(40,126,54,125)(41,135,47,116)(42,131,48,117)(43,132,49,118)(44,133,50,119)(45,134,46,120)(56,147,75,141)(57,148,71,142)(58,149,72,143)(59,150,73,144)(60,146,74,145)(61,155,67,136)(62,151,68,137)(63,152,69,138)(64,153,70,139)(65,154,66,140), (1,136,21,141)(2,137,22,142)(3,138,23,143)(4,139,24,144)(5,140,25,145)(6,45,20,40)(7,41,16,36)(8,42,17,37)(9,43,18,38)(10,44,19,39)(11,51,157,48)(12,52,158,49)(13,53,159,50)(14,54,160,46)(15,55,156,47)(26,146,34,154)(27,147,35,155)(28,148,31,151)(29,149,32,152)(30,150,33,153)(56,101,61,96)(57,102,62,97)(58,103,63,98)(59,104,64,99)(60,105,65,100)(66,114,74,106)(67,115,75,107)(68,111,71,108)(69,112,72,109)(70,113,73,110)(76,127,81,135)(77,128,82,131)(78,129,83,132)(79,130,84,133)(80,126,85,134)(86,120,94,125)(87,116,95,121)(88,117,91,122)(89,118,92,123)(90,119,93,124), (1,96)(2,97)(3,98)(4,99)(5,100)(6,85)(7,81)(8,82)(9,83)(10,84)(11,91)(12,92)(13,93)(14,94)(15,95)(16,76)(17,77)(18,78)(19,79)(20,80)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,135)(37,131)(38,132)(39,133)(40,134)(41,127)(42,128)(43,129)(44,130)(45,126)(46,125)(47,121)(48,122)(49,123)(50,124)(51,117)(52,118)(53,119)(54,120)(55,116)(56,141)(57,142)(58,143)(59,144)(60,145)(61,136)(62,137)(63,138)(64,139)(65,140)(66,154)(67,155)(68,151)(69,152)(70,153)(71,148)(72,149)(73,150)(74,146)(75,147)(86,160)(87,156)(88,157)(89,158)(90,159) );
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),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,47,35,41),(2,48,31,42),(3,49,32,43),(4,50,33,44),(5,46,34,45),(6,145,160,146),(7,141,156,147),(8,142,157,148),(9,143,158,149),(10,144,159,150),(11,151,17,137),(12,152,18,138),(13,153,19,139),(14,154,20,140),(15,155,16,136),(21,55,27,36),(22,51,28,37),(23,52,29,38),(24,53,30,39),(25,54,26,40),(56,87,75,81),(57,88,71,82),(58,89,72,83),(59,90,73,84),(60,86,74,85),(61,95,67,76),(62,91,68,77),(63,92,69,78),(64,93,70,79),(65,94,66,80),(96,121,115,127),(97,122,111,128),(98,123,112,129),(99,124,113,130),(100,125,114,126),(101,116,107,135),(102,117,108,131),(103,118,109,132),(104,119,110,133),(105,120,106,134)], [(1,107,35,101),(2,108,31,102),(3,109,32,103),(4,110,33,104),(5,106,34,105),(6,86,160,85),(7,87,156,81),(8,88,157,82),(9,89,158,83),(10,90,159,84),(11,77,17,91),(12,78,18,92),(13,79,19,93),(14,80,20,94),(15,76,16,95),(21,115,27,96),(22,111,28,97),(23,112,29,98),(24,113,30,99),(25,114,26,100),(36,127,55,121),(37,128,51,122),(38,129,52,123),(39,130,53,124),(40,126,54,125),(41,135,47,116),(42,131,48,117),(43,132,49,118),(44,133,50,119),(45,134,46,120),(56,147,75,141),(57,148,71,142),(58,149,72,143),(59,150,73,144),(60,146,74,145),(61,155,67,136),(62,151,68,137),(63,152,69,138),(64,153,70,139),(65,154,66,140)], [(1,136,21,141),(2,137,22,142),(3,138,23,143),(4,139,24,144),(5,140,25,145),(6,45,20,40),(7,41,16,36),(8,42,17,37),(9,43,18,38),(10,44,19,39),(11,51,157,48),(12,52,158,49),(13,53,159,50),(14,54,160,46),(15,55,156,47),(26,146,34,154),(27,147,35,155),(28,148,31,151),(29,149,32,152),(30,150,33,153),(56,101,61,96),(57,102,62,97),(58,103,63,98),(59,104,64,99),(60,105,65,100),(66,114,74,106),(67,115,75,107),(68,111,71,108),(69,112,72,109),(70,113,73,110),(76,127,81,135),(77,128,82,131),(78,129,83,132),(79,130,84,133),(80,126,85,134),(86,120,94,125),(87,116,95,121),(88,117,91,122),(89,118,92,123),(90,119,93,124)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,85),(7,81),(8,82),(9,83),(10,84),(11,91),(12,92),(13,93),(14,94),(15,95),(16,76),(17,77),(18,78),(19,79),(20,80),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,135),(37,131),(38,132),(39,133),(40,134),(41,127),(42,128),(43,129),(44,130),(45,126),(46,125),(47,121),(48,122),(49,123),(50,124),(51,117),(52,118),(53,119),(54,120),(55,116),(56,141),(57,142),(58,143),(59,144),(60,145),(61,136),(62,137),(63,138),(64,139),(65,140),(66,154),(67,155),(68,151),(69,152),(70,153),(71,148),(72,149),(73,150),(74,146),(75,147),(86,160),(87,156),(88,157),(89,158),(90,159)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 29 | 39 |
| 0 | 0 | 11 | 12 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 15 | 23 |
| 0 | 0 | 8 | 26 |
| 40 | 2 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 15 | 23 |
| 0 | 0 | 17 | 26 |
| 1 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 15 | 23 |
| 0 | 0 | 17 | 26 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,29,11,0,0,39,12],[40,0,0,0,0,40,0,0,0,0,15,8,0,0,23,26],[40,40,0,0,2,1,0,0,0,0,15,17,0,0,23,26],[1,1,0,0,0,40,0,0,0,0,15,17,0,0,23,26] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AF | 20A | ··· | 20AN | 20AO | ··· | 20BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2- (1+4) | C5×2- (1+4) |
| kernel | C5×Q8⋊5D4 | D4×C20 | Q8×C20 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C4.4D4 | Q8×C2×C10 | C10×C4○D4 | Q8⋊5D4 | C4×D4 | C4×Q8 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C22×Q8 | C2×C4○D4 | C5×Q8 | C2×C10 | Q8 | C22 | C10 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 4 | 12 | 4 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_5D_4
% in TeX
G:=Group("C5xQ8:5D4"); // GroupNames label
G:=SmallGroup(320,1550);
// by ID
G=gap.SmallGroup(320,1550);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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