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G = C2×Q64order 128 = 27

Direct product of C2 and Q64

direct product, p-group, metabelian, nilpotent (class 5), monomial

Aliases: C2×Q64, C8.20D8, C4.8D16, C16.11D4, C16.8C23, C32.5C22, C22.16D16, Q32.1C22, (C2×C32).4C2, (C2×C4).90D8, C4.15(C2×D8), C8.47(C2×D4), C2.14(C2×D16), (C2×C8).259D4, (C2×Q32).4C2, (C2×C16).90C22, SmallGroup(128,993)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C2×Q64
C1C2C4C8C16C2×C16C2×Q32 — C2×Q64
C1C2C4C8C16 — C2×Q64
C1C22C2×C4C2×C8C2×C16 — C2×Q64
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2×Q64

Generators and relations for C2×Q64
 G = < a,b,c | a2=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >

8C4
8C4
8C4
8C4
4Q8
4Q8
4Q8
4Q8
8C2×C4
8Q8
8C2×C4
8Q8
2Q16
2Q16
2Q16
2Q16
4C2×Q8
4Q16
4Q16
4C2×Q8
2C2×Q16
2Q32
2C2×Q16
2Q32

Smallest permutation representation of C2×Q64
Regular action on 128 points
Generators in S128
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 127)(52 128)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)
(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 99 17 115)(2 98 18 114)(3 97 19 113)(4 128 20 112)(5 127 21 111)(6 126 22 110)(7 125 23 109)(8 124 24 108)(9 123 25 107)(10 122 26 106)(11 121 27 105)(12 120 28 104)(13 119 29 103)(14 118 30 102)(15 117 31 101)(16 116 32 100)(33 65 49 81)(34 96 50 80)(35 95 51 79)(36 94 52 78)(37 93 53 77)(38 92 54 76)(39 91 55 75)(40 90 56 74)(41 89 57 73)(42 88 58 72)(43 87 59 71)(44 86 60 70)(45 85 61 69)(46 84 62 68)(47 83 63 67)(48 82 64 66)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D16A···16H32A···32P
order1222444444888816···1632···32
size1111221616161622222···22···2

38 irreducible representations

dim11112222222
type++++++++++-
imageC1C2C2C2D4D4D8D8D16D16Q64
kernelC2×Q64C2×C32Q64C2×Q32C16C2×C8C8C2×C4C4C22C2
# reps114211224416

Matrix representation of C2×Q64 in GL3(𝔽97) generated by

9600
0960
0096
,
9600
01691
0616
,
9600
06966
06628
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[96,0,0,0,16,6,0,91,16],[96,0,0,0,69,66,0,66,28] >;

C2×Q64 in GAP, Magma, Sage, TeX

C_2\times Q_{64}
% in TeX

G:=Group("C2xQ64");
// GroupNames label

G:=SmallGroup(128,993);
// by ID

G=gap.SmallGroup(128,993);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,448,141,456,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C2×Q64 in TeX

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