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G = C42.7Dic5order 320 = 26·5

4th non-split extension by C42 of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.7Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C20.55D4 — C42.7Dic5
 Lower central C5 — C2×C10 — C42.7Dic5
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.7Dic5
G = < a,b,c,d | a4=b4=1, c10=a2, d2=c5, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 222 in 110 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C52C8, C2×C20, C2×C20, C22×C10, C42.6C4, C2×C52C8, C4×C20, C22×C20, C42.D5, C203C8, C20.55D4, C2×C4×C20, C42.7Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C4○D4, Dic5, D10, C42⋊C2, C2×M4(2), C2×Dic5, C22×D5, C42.6C4, C4.Dic5, C4○D20, C22×Dic5, C2×C4.Dic5, C23.21D10, C42.7Dic5

Smallest permutation representation of C42.7Dic5
On 160 points
Generators in S160
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 92 71 82)(62 93 72 83)(63 94 73 84)(64 95 74 85)(65 96 75 86)(66 97 76 87)(67 98 77 88)(68 99 78 89)(69 100 79 90)(70 81 80 91)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 156 131 146)(122 157 132 147)(123 158 133 148)(124 159 134 149)(125 160 135 150)(126 141 136 151)(127 142 137 152)(128 143 138 153)(129 144 139 154)(130 145 140 155)
(1 105 34 49)(2 106 35 50)(3 107 36 51)(4 108 37 52)(5 109 38 53)(6 110 39 54)(7 111 40 55)(8 112 21 56)(9 113 22 57)(10 114 23 58)(11 115 24 59)(12 116 25 60)(13 117 26 41)(14 118 27 42)(15 119 28 43)(16 120 29 44)(17 101 30 45)(18 102 31 46)(19 103 32 47)(20 104 33 48)(61 147 97 127)(62 148 98 128)(63 149 99 129)(64 150 100 130)(65 151 81 131)(66 152 82 132)(67 153 83 133)(68 154 84 134)(69 155 85 135)(70 156 86 136)(71 157 87 137)(72 158 88 138)(73 159 89 139)(74 160 90 140)(75 141 91 121)(76 142 92 122)(77 143 93 123)(78 144 94 124)(79 145 95 125)(80 146 96 126)
(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 86 6 91 11 96 16 81)(2 95 7 100 12 85 17 90)(3 84 8 89 13 94 18 99)(4 93 9 98 14 83 19 88)(5 82 10 87 15 92 20 97)(21 73 26 78 31 63 36 68)(22 62 27 67 32 72 37 77)(23 71 28 76 33 61 38 66)(24 80 29 65 34 70 39 75)(25 69 30 74 35 79 40 64)(41 154 46 159 51 144 56 149)(42 143 47 148 52 153 57 158)(43 152 48 157 53 142 58 147)(44 141 49 146 54 151 59 156)(45 150 50 155 55 160 60 145)(101 130 106 135 111 140 116 125)(102 139 107 124 112 129 117 134)(103 128 108 133 113 138 118 123)(104 137 109 122 114 127 119 132)(105 126 110 131 115 136 120 121)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 2 20 ··· 20 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 D5 M4(2) C4○D4 M4(2) Dic5 D10 Dic5 D10 C4.Dic5 C4○D20 C4.Dic5 kernel C42.7Dic5 C42.D5 C20⋊3C8 C20.55D4 C2×C4×C20 C4×C20 C22×C20 C2×C42 C20 C20 C2×C10 C42 C42 C22×C4 C22×C4 C4 C4 C22 # reps 1 2 2 2 1 4 4 2 4 4 4 4 4 4 2 16 16 16

Matrix representation of C42.7Dic5 in GL4(𝔽41) generated by

 32 0 0 0 0 9 0 0 0 0 40 0 0 0 38 1
,
 1 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 32 0 0 0 0 32 0 0 0 0 31 0 0 0 20 4
,
 0 1 0 0 32 0 0 0 0 0 14 18 0 0 5 27
G:=sub<GL(4,GF(41))| [32,0,0,0,0,9,0,0,0,0,40,38,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,31,20,0,0,0,4],[0,32,0,0,1,0,0,0,0,0,14,5,0,0,18,27] >;

C42.7Dic5 in GAP, Magma, Sage, TeX

C_4^2._7{\rm Dic}_5
% in TeX

G:=Group("C4^2.7Dic5");
// GroupNames label

G:=SmallGroup(320,553);
// by ID

G=gap.SmallGroup(320,553);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,100,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=c^5,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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