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## G = (C5×Q8).D4order 320 = 26·5

### 6th non-split extension by C5×Q8 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×Q8).D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Q8×Dic5 — (C5×Q8).D4
 Lower central C5 — C10 — C2×C20 — (C5×Q8).D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for (C5×Q8).D4
G = < a,b,c,d,e | a5=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 390 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, Q8.D4, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C5⋊Q16, C23.D5, C2×C40, C5×SD16, C2×Dic10, D4×C10, Q8×C10, C20.8Q8, C20.44D4, D4⋊Dic5, C20.17D4, C2×C5⋊Q16, Q8×Dic5, C10×SD16, (C5×Q8).D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, C5⋊D4, C22×D5, Q8.D4, D4×D5, D42D5, C2×C5⋊D4, SD16⋊D5, SD163D5, Dic5⋊D4, (C5×Q8).D4

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 C8.C22 D4⋊2D5 D4×D5 SD16⋊D5 SD16⋊3D5 kernel (C5×Q8).D4 C20.8Q8 C20.44D4 D4⋊Dic5 C20.17D4 C2×C5⋊Q16 Q8×Dic5 C10×SD16 C2×Dic5 C5×Q8 C2×SD16 C20 C2×C8 C2×D4 C2×Q8 C10 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

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

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 14 10
,
 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 26 15 0 0 15 15 0 0 0 0 40 0 0 0 0 40
,
 9 0 0 0 0 9 0 0 0 0 9 9 0 0 0 32
,
 9 0 0 0 0 32 0 0 0 0 9 9 0 0 23 32
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,14,0,0,0,10],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[26,15,0,0,15,15,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,9,32],[9,0,0,0,0,32,0,0,0,0,9,23,0,0,9,32] >;

(C5×Q8).D4 in GAP, Magma, Sage, TeX

(C_5\times Q_8).D_4
% in TeX

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

G:=SmallGroup(320,793);
// by ID

G=gap.SmallGroup(320,793);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,1094,135,570,297,136,12550]);
// Polycyclic

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

׿
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𝔽