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G = C4⋊C4.231D10order 320 = 26·5

9th non-split extension by C4⋊C4 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.231D10, (C2×C20).286D4, C4.89(C4○D20), C10.Q1626C2, C20.Q826C2, C10.50(C2×SD16), (C2×C10).33SD16, (C22×C4).97D10, C20.177(C4○D4), (C2×C20).324C23, C20.55D4.4C2, (C22×C10).189D4, C23.80(C5⋊D4), C54(C23.47D4), C22.8(D4.D5), C20.48D4.10C2, C2.7(C20.C23), C10.85(C8.C22), C4⋊Dic5.133C22, (C22×C20).139C22, (C2×Dic10).101C22, C10.61(C22.D4), C2.11(C23.23D10), (C2×C4⋊C4).9D5, (C10×C4⋊C4).8C2, C2.5(C2×D4.D5), (C2×C10).444(C2×D4), (C2×C4).34(C5⋊D4), (C5×C4⋊C4).262C22, (C2×C52C8).84C22, (C2×C4).424(C22×D5), C22.133(C2×C5⋊D4), SmallGroup(320,598)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊C4.231D10
C1C5C10C20C2×C20C2×Dic10C20.48D4 — C4⋊C4.231D10
C5C10C2×C20 — C4⋊C4.231D10
C1C22C22×C4C2×C4⋊C4

Generators and relations for C4⋊C4.231D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=b2c-1 >

Subgroups: 318 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×6], C22×C10, C23.47D4, C2×C52C8 [×2], C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×C20, C22×C20, C20.Q8 [×2], C10.Q16 [×2], C20.55D4, C20.48D4, C10×C4⋊C4, C4⋊C4.231D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8.C22, C5⋊D4 [×2], C22×D5, C23.47D4, D4.D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, C2×D4.D5, C20.C23, C4⋊C4.231D10

Smallest permutation representation of C4⋊C4.231D10
On 160 points
Generators in S160
(1 36 8 31)(2 37 9 32)(3 38 10 33)(4 39 6 34)(5 40 7 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)(81 113 105 96)(82 114 106 97)(83 115 107 98)(84 116 108 99)(85 117 109 100)(86 118 110 91)(87 119 101 92)(88 120 102 93)(89 111 103 94)(90 112 104 95)(121 160 148 135)(122 151 149 136)(123 152 150 137)(124 153 141 138)(125 154 142 139)(126 155 143 140)(127 156 144 131)(128 157 145 132)(129 158 146 133)(130 159 147 134)
(1 51 13 41)(2 52 14 42)(3 53 15 43)(4 54 11 44)(5 55 12 45)(6 59 16 49)(7 60 17 50)(8 56 18 46)(9 57 19 47)(10 58 20 48)(21 74 34 64)(22 75 35 65)(23 71 31 61)(24 72 32 62)(25 73 33 63)(26 79 39 69)(27 80 40 70)(28 76 36 66)(29 77 37 67)(30 78 38 68)(81 158 86 153)(82 159 87 154)(83 160 88 155)(84 151 89 156)(85 152 90 157)(91 141 96 146)(92 142 97 147)(93 143 98 148)(94 144 99 149)(95 145 100 150)(101 139 106 134)(102 140 107 135)(103 131 108 136)(104 132 109 137)(105 133 110 138)(111 127 116 122)(112 128 117 123)(113 129 118 124)(114 130 119 125)(115 121 120 126)
(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)
(1 104 18 85)(2 108 19 89)(3 102 20 83)(4 106 16 87)(5 110 17 81)(6 82 11 101)(7 86 12 105)(8 90 13 109)(9 84 14 103)(10 88 15 107)(21 92 39 114)(22 96 40 118)(23 100 36 112)(24 94 37 116)(25 98 38 120)(26 119 34 97)(27 113 35 91)(28 117 31 95)(29 111 32 99)(30 115 33 93)(41 123 56 145)(42 127 57 149)(43 121 58 143)(44 125 59 147)(45 129 60 141)(46 150 51 128)(47 144 52 122)(48 148 53 126)(49 142 54 130)(50 146 55 124)(61 132 76 152)(62 136 77 156)(63 140 78 160)(64 134 79 154)(65 138 80 158)(66 157 71 137)(67 151 72 131)(68 155 73 135)(69 159 74 139)(70 153 75 133)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I5A5B8A8B8C8D10A···10N20A···20X
order122222444···44455888810···1020···20
size111122224···4404022202020202···24···4

59 irreducible representations

dim1111112222222222444
type+++++++++++--
imageC1C2C2C2C2C2D4D4D5C4○D4SD16D10D10C5⋊D4C5⋊D4C4○D20C8.C22D4.D5C20.C23
kernelC4⋊C4.231D10C20.Q8C10.Q16C20.55D4C20.48D4C10×C4⋊C4C2×C20C22×C10C2×C4⋊C4C20C2×C10C4⋊C4C22×C4C2×C4C23C4C10C22C2
# reps12211111244424416144

Matrix representation of C4⋊C4.231D10 in GL6(𝔽41)

100000
010000
001000
000100
0000117
00002440
,
900000
090000
0040000
0004000
0000171
00004024
,
1140000
0400000
0025000
00332300
000010
000001
,
2740000
2140000
0031100
0033800
000090
00001132

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,17,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,40,0,0,0,0,1,24],[1,0,0,0,0,0,14,40,0,0,0,0,0,0,25,33,0,0,0,0,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,2,0,0,0,0,4,14,0,0,0,0,0,0,3,3,0,0,0,0,11,38,0,0,0,0,0,0,9,11,0,0,0,0,0,32] >;

C4⋊C4.231D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{231}D_{10}
% in TeX

G:=Group("C4:C4.231D10");
// GroupNames label

G:=SmallGroup(320,598);
// by ID

G=gap.SmallGroup(320,598);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,100,1123,297,136,12550]);
// Polycyclic

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

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