Copied to
clipboard

G = C85(C4⋊C4)  order 128 = 27

2nd semidirect product of C8 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

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

Aliases: C85(C4⋊C4), C2.D87C4, C4.7(C4×Q8), C2.11(C4×D8), (C2×C4).58D8, (C2×C8).44Q8, (C2×C8).348D4, C2.11(C4×Q16), (C2×C4).30Q16, C2.4(C87D4), C2.2(C82Q8), C22.44(C2×D8), (C22×C4).556D4, C22.175(C4×D4), C23.789(C2×D4), C22.32(C4⋊Q8), C4.75(C22⋊Q8), C4.5(C42.C2), C2.3(C8.5Q8), C22.37(C2×Q16), C2.4(C8.18D4), C22.72(C4○D8), C22.4Q16.15C2, (C22×C8).487C22, C22.135(C4⋊D4), (C2×C42).1068C22, (C22×C4).1391C23, C23.65C23.11C2, C2.9(C23.65C23), (C2×C4×C8).35C2, C4.40(C2×C4⋊C4), C4⋊C4.88(C2×C4), (C2×C2.D8).8C2, (C2×C8).171(C2×C4), (C2×C4).203(C2×Q8), (C2×C4).1348(C2×D4), (C2×C4⋊C4).75C22, (C2×C4).586(C4○D4), (C2×C4).409(C22×C4), SmallGroup(128,674)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C85(C4⋊C4)
C1C2C22C2×C4C22×C4C22×C8C2×C4×C8 — C85(C4⋊C4)
C1C2C2×C4 — C85(C4⋊C4)
C1C23C2×C42 — C85(C4⋊C4)
C1C2C2C22×C4 — C85(C4⋊C4)

Generators and relations for C85(C4⋊C4)
 G = < a,b,c | a8=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 228 in 124 conjugacy classes, 68 normal (36 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], C23, C42 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C2.D8 [×4], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C22.4Q16 [×2], C23.65C23 [×2], C2×C4×C8, C2×C2.D8 [×2], C85(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×D8, C2×Q16, C4○D8 [×2], C23.65C23, C4×D8, C4×Q16, C87D4, C8.18D4, C8.5Q8, C82Q8, C85(C4⋊C4)

Smallest permutation representation of C85(C4⋊C4)
Regular action on 128 points
Generators in S128
(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)
(1 104 19 118)(2 103 20 117)(3 102 21 116)(4 101 22 115)(5 100 23 114)(6 99 24 113)(7 98 17 120)(8 97 18 119)(9 27 46 124)(10 26 47 123)(11 25 48 122)(12 32 41 121)(13 31 42 128)(14 30 43 127)(15 29 44 126)(16 28 45 125)(33 85 51 71)(34 84 52 70)(35 83 53 69)(36 82 54 68)(37 81 55 67)(38 88 56 66)(39 87 49 65)(40 86 50 72)(57 105 75 95)(58 112 76 94)(59 111 77 93)(60 110 78 92)(61 109 79 91)(62 108 80 90)(63 107 73 89)(64 106 74 96)
(1 50 45 77)(2 51 46 78)(3 52 47 79)(4 53 48 80)(5 54 41 73)(6 55 42 74)(7 56 43 75)(8 49 44 76)(9 60 20 33)(10 61 21 34)(11 62 22 35)(12 63 23 36)(13 64 24 37)(14 57 17 38)(15 58 18 39)(16 59 19 40)(25 90 115 69)(26 91 116 70)(27 92 117 71)(28 93 118 72)(29 94 119 65)(30 95 120 66)(31 96 113 67)(32 89 114 68)(81 128 106 99)(82 121 107 100)(83 122 108 101)(84 123 109 102)(85 124 110 103)(86 125 111 104)(87 126 112 97)(88 127 105 98)

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

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

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

44 conjugacy classes

class 1 2A···2G4A···4L4M···4T8A···8P
order12···24···44···48···8
size11···12···28···82···2

44 irreducible representations

dim1111112222222
type++++++-++-
imageC1C2C2C2C2C4D4Q8D4D8Q16C4○D4C4○D8
kernelC85(C4⋊C4)C22.4Q16C23.65C23C2×C4×C8C2×C2.D8C2.D8C2×C8C2×C8C22×C4C2×C4C2×C4C2×C4C22
# reps1221282424448

Matrix representation of C85(C4⋊C4) in GL5(𝔽17)

160000
016200
016100
000143
0001414
,
160000
013800
00400
000016
000160
,
130000
04900
041300
00010
00001

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,14,14,0,0,0,3,14],[16,0,0,0,0,0,13,0,0,0,0,8,4,0,0,0,0,0,0,16,0,0,0,16,0],[13,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,1,0,0,0,0,0,1] >;

C85(C4⋊C4) in GAP, Magma, Sage, TeX

C_8\rtimes_5(C_4\rtimes C_4)
% in TeX

G:=Group("C8:5(C4:C4)");
// GroupNames label

G:=SmallGroup(128,674);
// by ID

G=gap.SmallGroup(128,674);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,436,2804,172]);
// Polycyclic

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

׿
×
𝔽