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## G = (C5×Q8)⋊13D4order 320 = 26·5

### 1st semidirect product of C5×Q8 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×Q8)⋊13D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊7D4 — (C5×Q8)⋊13D4
 Lower central C5 — C10 — C2×C20 — (C5×Q8)⋊13D4
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C5×Q8)⋊13D4
G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=dbd-1=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >

Subgroups: 558 in 158 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×4], Q8 [×6], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×5], Dic5, C20 [×2], C20 [×5], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×8], C5×Q8 [×4], C5×Q8 [×6], C22×D5, C22×C10, Q8⋊D4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, Q8⋊D5 [×4], C2×D20, C2×C5⋊D4, C22×C20, C22×C20, Q8×C10 [×2], Q8×C10 [×5], C20.55D4, Q8⋊Dic5 [×2], C207D4, C2×Q8⋊D5 [×2], Q8×C2×C10, (C5×Q8)⋊13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, C5⋊D4 [×6], C22×D5, Q8⋊D4, Q8⋊D5 [×2], C2×C5⋊D4 [×3], C2×Q8⋊D5, C20.C23, C242D5, (C5×Q8)⋊13D4

Smallest permutation representation of (C5×Q8)⋊13D4
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 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)(81 116 86 111)(82 117 87 112)(83 118 88 113)(84 119 89 114)(85 120 90 115)(91 106 96 101)(92 107 97 102)(93 108 98 103)(94 109 99 104)(95 110 100 105)(121 151 126 156)(122 152 127 157)(123 153 128 158)(124 154 129 159)(125 155 130 160)(131 141 136 146)(132 142 137 147)(133 143 138 148)(134 144 139 149)(135 145 140 150)
(1 126 6 121)(2 127 7 122)(3 128 8 123)(4 129 9 124)(5 130 10 125)(11 136 16 131)(12 137 17 132)(13 138 18 133)(14 139 19 134)(15 140 20 135)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(41 81 46 86)(42 82 47 87)(43 83 48 88)(44 84 49 89)(45 85 50 90)(51 91 56 96)(52 92 57 97)(53 93 58 98)(54 94 59 99)(55 95 60 100)(61 101 66 106)(62 102 67 107)(63 103 68 108)(64 104 69 109)(65 105 70 110)(71 111 76 116)(72 112 77 117)(73 113 78 118)(74 114 79 119)(75 115 80 120)
(1 81 11 91)(2 85 12 95)(3 84 13 94)(4 83 14 93)(5 82 15 92)(6 86 16 96)(7 90 17 100)(8 89 18 99)(9 88 19 98)(10 87 20 97)(21 106 31 116)(22 110 32 120)(23 109 33 119)(24 108 34 118)(25 107 35 117)(26 101 36 111)(27 105 37 115)(28 104 38 114)(29 103 39 113)(30 102 40 112)(41 141 51 151)(42 145 52 155)(43 144 53 154)(44 143 54 153)(45 142 55 152)(46 146 56 156)(47 150 57 160)(48 149 58 159)(49 148 59 158)(50 147 60 157)(61 121 71 131)(62 125 72 135)(63 124 73 134)(64 123 74 133)(65 122 75 132)(66 126 76 136)(67 130 77 140)(68 129 78 139)(69 128 79 138)(70 127 80 137)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 61)(42 65)(43 64)(44 63)(45 62)(46 66)(47 70)(48 69)(49 68)(50 67)(51 71)(52 75)(53 74)(54 73)(55 72)(56 76)(57 80)(58 79)(59 78)(60 77)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 116)(102 120)(103 119)(104 118)(105 117)(106 111)(107 115)(108 114)(109 113)(110 112)(121 151)(122 155)(123 154)(124 153)(125 152)(126 156)(127 160)(128 159)(129 158)(130 157)(131 141)(132 145)(133 144)(134 143)(135 142)(136 146)(137 150)(138 149)(139 148)(140 147)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 40 2 2 4 ··· 4 40 2 2 20 20 20 20 2 ··· 2 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 D4 D5 SD16 D10 D10 C5⋊D4 C5⋊D4 C5⋊D4 C8.C22 Q8⋊D5 C20.C23 kernel (C5×Q8)⋊13D4 C20.55D4 Q8⋊Dic5 C20⋊7D4 C2×Q8⋊D5 Q8×C2×C10 C2×C20 C5×Q8 C22×C10 C22×Q8 C2×C10 C22×C4 C2×Q8 C2×C4 Q8 C23 C10 C22 C2 # reps 1 1 2 1 2 1 1 4 1 2 4 2 4 4 16 4 1 4 4

Matrix representation of (C5×Q8)⋊13D4 in GL4(𝔽41) generated by

 34 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 9 0 0 18 40
,
 17 40 0 0 1 24 0 0 0 0 11 29 0 0 17 30
,
 24 1 0 0 38 17 0 0 0 0 1 0 0 0 18 40
,
 1 0 0 0 34 40 0 0 0 0 1 0 0 0 18 40
G:=sub<GL(4,GF(41))| [34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,18,0,0,9,40],[17,1,0,0,40,24,0,0,0,0,11,17,0,0,29,30],[24,38,0,0,1,17,0,0,0,0,1,18,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,1,18,0,0,0,40] >;

(C5×Q8)⋊13D4 in GAP, Magma, Sage, TeX

(C_5\times Q_8)\rtimes_{13}D_4
% in TeX

G:=Group("(C5xQ8):13D4");
// GroupNames label

G:=SmallGroup(320,854);
// by ID

G=gap.SmallGroup(320,854);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,184,1123,297,136,12550]);
// 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,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
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