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G = C4⋊C4.178D10order 320 = 26·5

51st non-split extension by C4⋊C4 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.178D10, (D4×Dic5)⋊16C2, C4⋊D4.10D5, (C2×D4).152D10, C22⋊C4.47D10, C4.Dic1018C2, Dic53Q821C2, C20.201(C4○D4), C20.17D415C2, C4.67(D42D5), C20.48D431C2, (C2×C10).144C24, (C2×C20).501C23, (C22×C4).367D10, C23.11(C22×D5), Dic5.72(C4○D4), (D4×C10).118C22, C23.11D104C2, C23.D1014C2, C22.5(D42D5), C23.18D107C2, C4⋊Dic5.205C22, (C22×C10).15C23, (C4×Dic5).99C22, C22.165(C23×D5), C23.D5.21C22, (C22×C20).238C22, C56(C23.36C23), (C2×Dic5).236C23, C10.D4.15C22, (C2×Dic10).158C22, (C22×Dic5).105C22, (C2×C4×Dic5)⋊8C2, C2.35(D5×C4○D4), (C5×C4⋊D4).7C2, C10.149(C2×C4○D4), C2.32(C2×D42D5), (C2×C10).20(C4○D4), (C5×C4⋊C4).140C22, (C2×C4).292(C22×D5), (C5×C22⋊C4).9C22, SmallGroup(320,1272)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4⋊C4.178D10
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C4⋊C4.178D10
C5C2×C10 — C4⋊C4.178D10
C1C22C4⋊D4

Generators and relations for C4⋊C4.178D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 670 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C23 [×2], C10 [×3], C10 [×4], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×C10, C22×C10 [×2], C23.36C23, C4×Dic5 [×4], C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, D4×C10, D4×C10 [×2], C23.11D10 [×2], C23.D10 [×2], Dic53Q8, C4.Dic10, C2×C4×Dic5, C20.48D4, D4×Dic5, D4×Dic5 [×2], C23.18D10 [×2], C20.17D4, C5×C4⋊D4, C4⋊C4.178D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D42D5 [×4], C23×D5, C2×D42D5 [×2], D5×C4○D4, C4⋊C4.178D10

Smallest permutation representation of C4⋊C4.178D10
On 160 points
Generators in S160
(1 23 32 46)(2 24 33 47)(3 25 34 48)(4 26 35 49)(5 27 36 50)(6 28 37 41)(7 29 38 42)(8 30 39 43)(9 21 40 44)(10 22 31 45)(11 74 54 112)(12 75 55 113)(13 76 56 114)(14 77 57 115)(15 78 58 116)(16 79 59 117)(17 80 60 118)(18 71 51 119)(19 72 52 120)(20 73 53 111)(61 96 158 148)(62 97 159 149)(63 98 160 150)(64 99 151 141)(65 100 152 142)(66 91 153 143)(67 92 154 144)(68 93 155 145)(69 94 156 146)(70 95 157 147)(81 123 133 106)(82 124 134 107)(83 125 135 108)(84 126 136 109)(85 127 137 110)(86 128 138 101)(87 129 139 102)(88 130 140 103)(89 121 131 104)(90 122 132 105)
(1 116 111 6)(2 7 112 117)(3 118 113 8)(4 9 114 119)(5 120 115 10)(11 59 24 42)(12 43 25 60)(13 51 26 44)(14 45 27 52)(15 53 28 46)(16 47 29 54)(17 55 30 48)(18 49 21 56)(19 57 22 50)(20 41 23 58)(31 36 72 77)(32 78 73 37)(33 38 74 79)(34 80 75 39)(35 40 76 71)(61 66 108 103)(62 104 109 67)(63 68 110 105)(64 106 101 69)(65 70 102 107)(81 138 94 141)(82 142 95 139)(83 140 96 143)(84 144 97 131)(85 132 98 145)(86 146 99 133)(87 134 100 147)(88 148 91 135)(89 136 92 149)(90 150 93 137)(121 126 154 159)(122 160 155 127)(123 128 156 151)(124 152 157 129)(125 130 158 153)
(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)
(1 156 73 106)(2 155 74 105)(3 154 75 104)(4 153 76 103)(5 152 77 102)(6 151 78 101)(7 160 79 110)(8 159 80 109)(9 158 71 108)(10 157 72 107)(11 132 47 93)(12 131 48 92)(13 140 49 91)(14 139 50 100)(15 138 41 99)(16 137 42 98)(17 136 43 97)(18 135 44 96)(19 134 45 95)(20 133 46 94)(21 148 51 83)(22 147 52 82)(23 146 53 81)(24 145 54 90)(25 144 55 89)(26 143 56 88)(27 142 57 87)(28 141 58 86)(29 150 59 85)(30 149 60 84)(31 70 120 124)(32 69 111 123)(33 68 112 122)(34 67 113 121)(35 66 114 130)(36 65 115 129)(37 64 116 128)(38 63 117 127)(39 62 118 126)(40 61 119 125)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K···4P4Q4R4S4T5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order1222222244444444444···444445510···10101010101010101020···2020202020
size11112244222244555510···1020202020222···2444488884···48888

56 irreducible representations

dim1111111111122222222444
type++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4C4○D4D10D10D10D10D42D5D42D5D5×C4○D4
kernelC4⋊C4.178D10C23.11D10C23.D10Dic53Q8C4.Dic10C2×C4×Dic5C20.48D4D4×Dic5C23.18D10C20.17D4C5×C4⋊D4C4⋊D4Dic5C20C2×C10C22⋊C4C4⋊C4C22×C4C2×D4C4C22C2
# reps1221111321124444226444

Matrix representation of C4⋊C4.178D10 in GL6(𝔽41)

150000
16400000
0012500
00364000
000010
000001
,
940000
0320000
0012500
0004000
0000400
0000040
,
940000
21320000
001000
000100
000076
0000340
,
32370000
2090000
009000
000900
0000400
000081

G:=sub<GL(6,GF(41))| [1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,36,0,0,0,0,25,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,25,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,21,0,0,0,0,4,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,6,0],[32,20,0,0,0,0,37,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,8,0,0,0,0,0,1] >;

C4⋊C4.178D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{178}D_{10}
% in TeX

G:=Group("C4:C4.178D10");
// GroupNames label

G:=SmallGroup(320,1272);
// by ID

G=gap.SmallGroup(320,1272);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,794,297,12550]);
// Polycyclic

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

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