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G = C2.D85C4order 128 = 27

3rd semidirect product of C2.D8 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2.D85C4, C4⋊C4.14Q8, C4.26(C4×Q8), C2.8(C4×Q16), (C2×C4).48Q16, C2.5(D4.Q8), C2.8(D8⋊C4), C2.3(C4.Q16), C23.779(C2×D4), C22.159(C4×D4), (C22×C4).694D4, C22.34(C2×Q16), C22.64(C4○D8), (C22×C8).45C22, C4.14(C42.C2), C4.16(C422C2), C4.25(C42⋊C2), C22.83(C8⋊C22), C22.4Q16.10C2, (C2×C42).294C22, C22.78(C22⋊Q8), (C22×C4).1377C23, C2.4(C23.48D4), C2.5(C23.19D4), C22.7C42.21C2, C23.65C23.7C2, C22.91(C22.D4), C2.12(C23.63C23), (C4×C4⋊C4).17C2, C4⋊C4.79(C2×C4), (C2×C8).39(C2×C4), (C2×C2.D8).6C2, (C2×C4).272(C2×Q8), (C2×C4).574(C4○D4), (C2×C4⋊C4).772C22, (C2×C4).395(C22×C4), SmallGroup(128,653)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2.D85C4
C1C2C22C2×C4C22×C4C2×C4⋊C4C4×C4⋊C4 — C2.D85C4
C1C2C2×C4 — C2.D85C4
C1C23C2×C42 — C2.D85C4
C1C2C2C22×C4 — C2.D85C4

Generators and relations for C2.D85C4
 G = < a,b,c,d | a2=b8=d4=1, c2=a, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab5, dcd-1=ab4c >

Subgroups: 220 in 115 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], C23, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4, C22×C8 [×2], C22.7C42, C22.4Q16 [×3], C4×C4⋊C4, C23.65C23, C2×C2.D8, C2.D85C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Q16 [×2], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×Q16, C4○D8, C8⋊C22 [×2], C23.63C23, C4×Q16, D8⋊C4, C4.Q16, D4.Q8, C23.19D4, C23.48D4, C2.D85C4

Smallest permutation representation of C2.D85C4
Regular action on 128 points
Generators in S128
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(113 128)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)
(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)
(1 110 60 10)(2 109 61 9)(3 108 62 16)(4 107 63 15)(5 106 64 14)(6 105 57 13)(7 112 58 12)(8 111 59 11)(17 86 102 47)(18 85 103 46)(19 84 104 45)(20 83 97 44)(21 82 98 43)(22 81 99 42)(23 88 100 41)(24 87 101 48)(25 78 94 39)(26 77 95 38)(27 76 96 37)(28 75 89 36)(29 74 90 35)(30 73 91 34)(31 80 92 33)(32 79 93 40)(49 118 71 125)(50 117 72 124)(51 116 65 123)(52 115 66 122)(53 114 67 121)(54 113 68 128)(55 120 69 127)(56 119 70 126)
(1 78 127 103)(2 36 128 23)(3 80 121 97)(4 38 122 17)(5 74 123 99)(6 40 124 19)(7 76 125 101)(8 34 126 21)(9 32 68 84)(10 90 69 42)(11 26 70 86)(12 92 71 44)(13 28 72 88)(14 94 65 46)(15 30 66 82)(16 96 67 48)(18 60 39 120)(20 62 33 114)(22 64 35 116)(24 58 37 118)(25 51 85 106)(27 53 87 108)(29 55 81 110)(31 49 83 112)(41 105 89 50)(43 107 91 52)(45 109 93 54)(47 111 95 56)(57 79 117 104)(59 73 119 98)(61 75 113 100)(63 77 115 102)

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

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

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

38 conjugacy classes

class 1 2A···2G4A···4H4I···4R4S4T4U4V8A···8H
order12···24···44···444448···8
size11···12···24···488884···4

38 irreducible representations

dim1111111222224
type++++++-+-+
imageC1C2C2C2C2C2C4Q8D4Q16C4○D4C4○D8C8⋊C22
kernelC2.D85C4C22.7C42C22.4Q16C4×C4⋊C4C23.65C23C2×C2.D8C2.D8C4⋊C4C22×C4C2×C4C2×C4C22C22
# reps1131118224842

Matrix representation of C2.D85C4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
330000
3140000
000300
0011000
000020
000009
,
12120000
1250000
00111200
007600
000008
000020
,
010000
1600000
0013000
0001300
0000160
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,11,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,9],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,11,7,0,0,0,0,12,6,0,0,0,0,0,0,0,2,0,0,0,0,8,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C2.D85C4 in GAP, Magma, Sage, TeX

C_2.D_8\rtimes_5C_4
% in TeX

G:=Group("C2.D8:5C4");
// GroupNames label

G:=SmallGroup(128,653);
// by ID

G=gap.SmallGroup(128,653);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^4=1,c^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^5,d*c*d^-1=a*b^4*c>;
// generators/relations

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