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G = C4⋊C4.230D10order 320 = 26·5

8th non-split extension by C4⋊C4 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.230D10, (C2×C20).285D4, (C2×C10).17Q16, C10.33(C2×Q16), C4.88(C4○D20), C10.Q1625C2, C10.D826C2, (C22×C4).96D10, C20.176(C4○D4), C10.85(C8⋊C22), (C2×C20).323C23, C20.48D4.9C2, C20.55D4.3C2, (C22×C10).188D4, C23.79(C5⋊D4), C54(C23.48D4), C22.5(C5⋊Q16), C2.7(D4.D10), C4⋊Dic5.132C22, (C22×C20).138C22, (C2×Dic10).100C22, C10.60(C22.D4), C2.10(C23.23D10), (C2×C4⋊C4).8D5, (C10×C4⋊C4).7C2, C2.5(C2×C5⋊Q16), (C2×C10).443(C2×D4), (C2×C4).33(C5⋊D4), (C5×C4⋊C4).261C22, (C2×C52C8).83C22, (C2×C4).423(C22×D5), C22.132(C2×C5⋊D4), SmallGroup(320,597)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊C4.230D10
C1C5C10C20C2×C20C2×Dic10C20.48D4 — C4⋊C4.230D10
C5C10C2×C20 — C4⋊C4.230D10
C1C22C22×C4C2×C4⋊C4

Generators and relations for C4⋊C4.230D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-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], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×6], C22×C10, C23.48D4, C2×C52C8 [×2], C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×C20, C22×C20, C10.D8 [×2], C10.Q16 [×2], C20.55D4, C20.48D4, C10×C4⋊C4, C4⋊C4.230D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×Q16, C8⋊C22, C5⋊D4 [×2], C22×D5, C23.48D4, C5⋊Q16 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, D4.D10, C2×C5⋊Q16, C4⋊C4.230D10

Smallest permutation representation of C4⋊C4.230D10
On 160 points
Generators in S160
(1 31 8 36)(2 32 9 37)(3 33 10 38)(4 34 6 39)(5 35 7 40)(11 23 16 29)(12 24 17 30)(13 25 18 26)(14 21 19 27)(15 22 20 28)(41 76 46 71)(42 77 47 72)(43 78 48 73)(44 79 49 74)(45 80 50 75)(51 66 58 61)(52 67 59 62)(53 68 60 63)(54 69 56 64)(55 70 57 65)(81 96 105 115)(82 97 106 116)(83 98 107 117)(84 99 108 118)(85 100 109 119)(86 91 110 120)(87 92 101 111)(88 93 102 112)(89 94 103 113)(90 95 104 114)(121 132 153 143)(122 133 154 144)(123 134 155 145)(124 135 156 146)(125 136 157 147)(126 137 158 148)(127 138 159 149)(128 139 160 150)(129 140 151 141)(130 131 152 142)
(1 58 18 41)(2 59 19 42)(3 60 20 43)(4 56 16 44)(5 57 17 45)(6 54 11 49)(7 55 12 50)(8 51 13 46)(9 52 14 47)(10 53 15 48)(21 77 37 62)(22 78 38 63)(23 79 39 64)(24 80 40 65)(25 76 36 61)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)(81 128 110 155)(82 129 101 156)(83 130 102 157)(84 121 103 158)(85 122 104 159)(86 123 105 160)(87 124 106 151)(88 125 107 152)(89 126 108 153)(90 127 109 154)(91 145 115 139)(92 146 116 140)(93 147 117 131)(94 148 118 132)(95 149 119 133)(96 150 120 134)(97 141 111 135)(98 142 112 136)(99 143 113 137)(100 144 114 138)
(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 94 37 118)(22 98 38 112)(23 92 39 116)(24 96 40 120)(25 100 36 114)(26 119 31 95)(27 113 32 99)(28 117 33 93)(29 111 34 97)(30 115 35 91)(41 144 58 138)(42 148 59 132)(43 142 60 136)(44 146 56 140)(45 150 57 134)(46 133 51 149)(47 137 52 143)(48 131 53 147)(49 135 54 141)(50 139 55 145)(61 127 76 154)(62 121 77 158)(63 125 78 152)(64 129 79 156)(65 123 80 160)(66 159 71 122)(67 153 72 126)(68 157 73 130)(69 151 74 124)(70 155 75 128)

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

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

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

59 conjugacy classes

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

59 irreducible representations

dim1111112222222222444
type+++++++++-+++-
imageC1C2C2C2C2C2D4D4D5C4○D4Q16D10D10C5⋊D4C5⋊D4C4○D20C8⋊C22C5⋊Q16D4.D10
kernelC4⋊C4.230D10C10.D8C10.Q16C20.55D4C20.48D4C10×C4⋊C4C2×C20C22×C10C2×C4⋊C4C20C2×C10C4⋊C4C22×C4C2×C4C23C4C10C22C2
# reps12211111244424416144

Matrix representation of C4⋊C4.230D10 in GL4(𝔽41) generated by

40000
04000
00320
00149
,
32000
0900
00524
001636
,
10000
0400
0010
0001
,
03700
31000
002610
00215
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,14,0,0,0,9],[32,0,0,0,0,9,0,0,0,0,5,16,0,0,24,36],[10,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,31,0,0,37,0,0,0,0,0,26,2,0,0,10,15] >;

C4⋊C4.230D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{230}D_{10}
% in TeX

G:=Group("C4:C4.230D10");
// GroupNames label

G:=SmallGroup(320,597);
// by ID

G=gap.SmallGroup(320,597);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,268,1123,297,136,12550]);
// Polycyclic

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

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