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## G = (D4×C10).24C4order 320 = 26·5

### 5th non-split extension by D4×C10 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (D4×C10).24C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — (D4×C10).24C4
 Lower central C5 — C2×C10 — (D4×C10).24C4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (D4×C10).24C4
G = < a,b,c,d | a10=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a5b2c >

Subgroups: 350 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C52C8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, (C22×C8)⋊C2, C2×C52C8, C2×C52C8, C4.Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C20.55D4, C22×C52C8, C2×C4.Dic5, C10×C4○D4, (D4×C10).24C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C8○D4, C2×Dic5, C5⋊D4, C22×D5, (C22×C8)⋊C2, C23.D5, C22×Dic5, C2×C5⋊D4, D4.Dic5, C2×C23.D5, (D4×C10).24C4

Smallest permutation representation of (D4×C10).24C4
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 78 25 86)(2 79 26 87)(3 80 27 88)(4 71 28 89)(5 72 29 90)(6 73 30 81)(7 74 21 82)(8 75 22 83)(9 76 23 84)(10 77 24 85)(11 106 158 98)(12 107 159 99)(13 108 160 100)(14 109 151 91)(15 110 152 92)(16 101 153 93)(17 102 154 94)(18 103 155 95)(19 104 156 96)(20 105 157 97)(31 51 48 68)(32 52 49 69)(33 53 50 70)(34 54 41 61)(35 55 42 62)(36 56 43 63)(37 57 44 64)(38 58 45 65)(39 59 46 66)(40 60 47 67)(111 131 129 149)(112 132 130 150)(113 133 121 141)(114 134 122 142)(115 135 123 143)(116 136 124 144)(117 137 125 145)(118 138 126 146)(119 139 127 147)(120 140 128 148)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 71)(10 72)(11 106)(12 107)(13 108)(14 109)(15 110)(16 101)(17 102)(18 103)(19 104)(20 105)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 56)(32 57)(33 58)(34 59)(35 60)(36 51)(37 52)(38 53)(39 54)(40 55)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 131)(130 132)
(1 112 38 92 25 130 45 110)(2 111 39 91 26 129 46 109)(3 120 40 100 27 128 47 108)(4 119 31 99 28 127 48 107)(5 118 32 98 29 126 49 106)(6 117 33 97 30 125 50 105)(7 116 34 96 21 124 41 104)(8 115 35 95 22 123 42 103)(9 114 36 94 23 122 43 102)(10 113 37 93 24 121 44 101)(11 90 146 69 158 72 138 52)(12 89 147 68 159 71 139 51)(13 88 148 67 160 80 140 60)(14 87 149 66 151 79 131 59)(15 86 150 65 152 78 132 58)(16 85 141 64 153 77 133 57)(17 84 142 63 154 76 134 56)(18 83 143 62 155 75 135 55)(19 82 144 61 156 74 136 54)(20 81 145 70 157 73 137 53)

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 D5 D10 Dic5 Dic5 C8○D4 C5⋊D4 D4.Dic5 kernel (D4×C10).24C4 C20.55D4 C22×C5⋊2C8 C2×C4.Dic5 C10×C4○D4 D4×C10 Q8×C10 C2×C20 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C10 C2×C4 C2 # reps 1 4 1 1 1 6 2 4 2 6 6 2 8 16 8

Matrix representation of (D4×C10).24C4 in GL6(𝔽41)

 35 35 0 0 0 0 6 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 5 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 5 0 0 0 0 25 9 0 0 0 0 0 0 1 0 0 0 0 0 3 40
,
 15 31 0 0 0 0 39 26 0 0 0 0 0 0 38 0 0 0 0 0 0 38 0 0 0 0 0 0 38 2 0 0 0 0 36 3

G:=sub<GL(6,GF(41))| [35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,5,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,25,0,0,0,0,5,9,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[15,39,0,0,0,0,31,26,0,0,0,0,0,0,38,0,0,0,0,0,0,38,0,0,0,0,0,0,38,36,0,0,0,0,2,3] >;

(D4×C10).24C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})._{24}C_4
% in TeX

G:=Group("(D4xC10).24C4");
// GroupNames label

G:=SmallGroup(320,861);
// by ID

G=gap.SmallGroup(320,861);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^5*b^2*c>;
// generators/relations

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