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G = C40.37D4order 320 = 26·5

37th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.37D4, (C2×Q16)⋊9D5, C4.28(D4×D5), C408C410C2, (C2×C8).96D10, C52C8.10D4, C8.6(C5⋊D4), C55(C8.2D4), (C10×Q16)⋊10C2, C20.188(C2×D4), (C2×Q8).66D10, Dic5⋊Q86C2, (C2×Dic5).86D4, C22.281(D4×D5), C2.25(C20⋊D4), C10.34(C41D4), (C2×C40).151C22, (C2×C20).464C23, C20.23D4.8C2, (Q8×C10).93C22, C2.31(Q16⋊D5), (C2×D20).130C22, C10.81(C8.C22), (C4×Dic5).62C22, (C2×Dic10).137C22, C4.15(C2×C5⋊D4), (C2×Q8⋊D5).9C2, (C2×C5⋊Q16)⋊21C2, (C2×C40⋊C2).8C2, (C2×C10).375(C2×D4), (C2×C4).552(C22×D5), (C2×C52C8).168C22, SmallGroup(320,817)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.37D4
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C40.37D4
C5C10C2×C20 — C40.37D4
C1C22C2×C4C2×Q16

Generators and relations for C40.37D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=a20b-1 >

Subgroups: 526 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×6], C23, D5, C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], Q16 [×4], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16, C2×Q16, C52C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C8.2D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C2×Dic10, C2×D20, Q8×C10 [×2], C408C4, C2×C40⋊C2, C2×Q8⋊D5, C2×C5⋊Q16, Dic5⋊Q8, C20.23D4, C10×Q16, C40.37D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C8.2D4, D4×D5 [×2], C2×C5⋊D4, Q16⋊D5 [×2], C20⋊D4, C40.37D4

Smallest permutation representation of C40.37D4
On 160 points
Generators in S160
(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)
(1 113 137 53)(2 102 138 42)(3 91 139 71)(4 120 140 60)(5 109 141 49)(6 98 142 78)(7 87 143 67)(8 116 144 56)(9 105 145 45)(10 94 146 74)(11 83 147 63)(12 112 148 52)(13 101 149 41)(14 90 150 70)(15 119 151 59)(16 108 152 48)(17 97 153 77)(18 86 154 66)(19 115 155 55)(20 104 156 44)(21 93 157 73)(22 82 158 62)(23 111 159 51)(24 100 160 80)(25 89 121 69)(26 118 122 58)(27 107 123 47)(28 96 124 76)(29 85 125 65)(30 114 126 54)(31 103 127 43)(32 92 128 72)(33 81 129 61)(34 110 130 50)(35 99 131 79)(36 88 132 68)(37 117 133 57)(38 106 134 46)(39 95 135 75)(40 84 136 64)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 105)(42 84)(43 103)(44 82)(45 101)(46 120)(47 99)(48 118)(49 97)(50 116)(51 95)(52 114)(53 93)(54 112)(55 91)(56 110)(57 89)(58 108)(59 87)(60 106)(61 85)(62 104)(63 83)(64 102)(65 81)(66 100)(67 119)(68 98)(69 117)(70 96)(71 115)(72 94)(73 113)(74 92)(75 111)(76 90)(77 109)(78 88)(79 107)(80 86)(121 153)(122 132)(123 151)(124 130)(125 149)(126 128)(127 147)(129 145)(131 143)(133 141)(134 160)(135 139)(136 158)(138 156)(140 154)(142 152)(144 150)(146 148)(155 159)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444455888810···102020202020···2040···40
size1111402288202040224420202···244448···84···4

44 irreducible representations

dim1111111122222224444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10C5⋊D4C8.C22D4×D5D4×D5Q16⋊D5
kernelC40.37D4C408C4C2×C40⋊C2C2×Q8⋊D5C2×C5⋊Q16Dic5⋊Q8C20.23D4C10×Q16C52C8C40C2×Dic5C2×Q16C2×C8C2×Q8C8C10C4C22C2
# reps1111111122222482228

Matrix representation of C40.37D4 in GL8(𝔽41)

10000000
01000000
003570000
003500000
0000173677
00001018024
0000192133
00002921213
,
040000000
10000000
0035400000
003560000
000010370
00002104040
0000210400
000001400
,
10000000
040000000
00610000
006350000
00001000
0000214000
00000010
0000210040

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,17,10,19,29,0,0,0,0,36,18,21,21,0,0,0,0,7,0,3,21,0,0,0,0,7,24,3,3],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,1,21,21,0,0,0,0,0,0,0,0,1,0,0,0,0,37,40,40,40,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,21,0,21,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;

C40.37D4 in GAP, Magma, Sage, TeX

C_{40}._{37}D_4
% in TeX

G:=Group("C40.37D4");
// GroupNames label

G:=SmallGroup(320,817);
// by ID

G=gap.SmallGroup(320,817);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,184,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=a^20*b^-1>;
// generators/relations

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