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## G = C42.122D10order 320 = 26·5

### 122nd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.122D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×C4×D5 — C42⋊D5 — C42.122D10
 Lower central C5 — C2×C10 — C42.122D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.122D10
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=a2b-1, dcd=c9 >

Subgroups: 694 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.50C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C4×Dic10, C42⋊D5 [×2], C4.D20, C422D5 [×2], Dic53Q8, D208C4, D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], Dic5⋊Q8, C20.23D4, Q8×C20, C42.122D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.122D10

Smallest permutation representation of C42.122D10
On 160 points
Generators in S160
(1 67 129 30)(2 68 130 31)(3 69 131 32)(4 70 132 33)(5 71 133 34)(6 72 134 35)(7 73 135 36)(8 74 136 37)(9 75 137 38)(10 76 138 39)(11 77 139 40)(12 78 140 21)(13 79 121 22)(14 80 122 23)(15 61 123 24)(16 62 124 25)(17 63 125 26)(18 64 126 27)(19 65 127 28)(20 66 128 29)(41 142 112 86)(42 143 113 87)(43 144 114 88)(44 145 115 89)(45 146 116 90)(46 147 117 91)(47 148 118 92)(48 149 119 93)(49 150 120 94)(50 151 101 95)(51 152 102 96)(52 153 103 97)(53 154 104 98)(54 155 105 99)(55 156 106 100)(56 157 107 81)(57 158 108 82)(58 159 109 83)(59 160 110 84)(60 141 111 85)
(1 115 11 105)(2 55 12 45)(3 117 13 107)(4 57 14 47)(5 119 15 109)(6 59 16 49)(7 101 17 111)(8 41 18 51)(9 103 19 113)(10 43 20 53)(21 90 31 100)(22 157 32 147)(23 92 33 82)(24 159 34 149)(25 94 35 84)(26 141 36 151)(27 96 37 86)(28 143 38 153)(29 98 39 88)(30 145 40 155)(42 137 52 127)(44 139 54 129)(46 121 56 131)(48 123 58 133)(50 125 60 135)(61 83 71 93)(62 150 72 160)(63 85 73 95)(64 152 74 142)(65 87 75 97)(66 154 76 144)(67 89 77 99)(68 156 78 146)(69 91 79 81)(70 158 80 148)(102 136 112 126)(104 138 114 128)(106 140 116 130)(108 122 118 132)(110 124 120 134)
(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)
(1 114)(2 103)(3 112)(4 101)(5 110)(6 119)(7 108)(8 117)(9 106)(10 115)(11 104)(12 113)(13 102)(14 111)(15 120)(16 109)(17 118)(18 107)(19 116)(20 105)(21 153)(22 142)(23 151)(24 160)(25 149)(26 158)(27 147)(28 156)(29 145)(30 154)(31 143)(32 152)(33 141)(34 150)(35 159)(36 148)(37 157)(38 146)(39 155)(40 144)(41 131)(42 140)(43 129)(44 138)(45 127)(46 136)(47 125)(48 134)(49 123)(50 132)(51 121)(52 130)(53 139)(54 128)(55 137)(56 126)(57 135)(58 124)(59 133)(60 122)(61 84)(62 93)(63 82)(64 91)(65 100)(66 89)(67 98)(68 87)(69 96)(70 85)(71 94)(72 83)(73 92)(74 81)(75 90)(76 99)(77 88)(78 97)(79 86)(80 95)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 20 20 2 ··· 2 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 C4○D20 2- 1+4 Q8.10D10 D5×C4○D4 kernel C42.122D10 C4×Dic10 C42⋊D5 C4.D20 C42⋊2D5 Dic5⋊3Q8 D20⋊8C4 D10⋊Q8 C4⋊C4⋊D5 Dic5⋊Q8 C20.23D4 Q8×C20 C4×Q8 Dic5 C20 C42 C4⋊C4 C2×Q8 C4 C10 C2 C2 # reps 1 1 2 1 2 1 1 2 2 1 1 1 2 4 4 6 6 2 16 1 4 4

Matrix representation of C42.122D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 18 1
,
 1 6 0 0 0 0 35 6 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 4 5 0 0 0 0 38 37
,
 1 0 0 0 0 0 35 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 37 36 0 0 0 0 3 4

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,18,0,0,0,0,0,1],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,4,38,0,0,0,0,5,37],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,37,3,0,0,0,0,36,4] >;

C42.122D10 in GAP, Magma, Sage, TeX

C_4^2._{122}D_{10}
% in TeX

G:=Group("C4^2.122D10");
// GroupNames label

G:=SmallGroup(320,1240);
// by ID

G=gap.SmallGroup(320,1240);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,232,758,100,794,136,12550]);
// Polycyclic

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

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