Copied to
clipboard

G = C2×Q8.Q8order 128 = 27

Direct product of C2 and Q8.Q8

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

Aliases: C2×Q8.Q8, C42.218D4, C42.331C23, Q8.1(C2×Q8), (C2×Q8).25Q8, C4⋊C4.38C23, C4.26(C22×Q8), C4⋊C8.308C22, (C2×C8).138C23, (C2×C4).273C24, C23.869(C2×D4), (C22×C4).719D4, C4.66(C22⋊Q8), C22.94(C4○D8), (C2×Q8).363C23, (C4×Q8).295C22, C4.Q8.146C22, C2.D8.165C22, (C22×C8).344C22, (C2×C42).819C22, C22.533(C22×D4), (C22×C4).1543C23, Q8⋊C4.148C22, C22.101(C22⋊Q8), C42.C2.101C22, (C22×Q8).473C22, C22.108(C8.C22), (C2×C4⋊C8).46C2, (C2×C4×Q8).51C2, C2.18(C2×C4○D8), C4.83(C2×C4○D4), (C2×C4).480(C2×D4), (C2×C4).321(C2×Q8), (C2×C4.Q8).32C2, (C2×C2.D8).28C2, C2.54(C2×C22⋊Q8), C2.22(C2×C8.C22), (C2×C4).839(C4○D4), (C2×C4⋊C4).602C22, (C2×Q8⋊C4).30C2, (C2×C42.C2).31C2, SmallGroup(128,1807)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×Q8.Q8
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C2×Q8.Q8
C1C2C2×C4 — C2×Q8.Q8
C1C23C2×C42 — C2×Q8.Q8
C1C2C2C2×C4 — C2×Q8.Q8

Generators and relations for C2×Q8.Q8
 G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 300 in 192 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×14], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C4⋊C4 [×15], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×3], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×3], C2×C4⋊C4 [×3], C4×Q8 [×4], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22×Q8, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8, C2×C2.D8, Q8.Q8 [×8], C2×C4×Q8, C2×C42.C2, C2×Q8.Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C4○D8 [×2], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, Q8.Q8 [×4], C2×C22⋊Q8, C2×C4○D8, C2×C8.C22, C2×Q8.Q8

Smallest permutation representation of C2×Q8.Q8
Regular action on 128 points
Generators in S128
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)(17 31)(18 32)(19 29)(20 30)(21 27)(22 28)(23 25)(24 26)(33 47)(34 48)(35 45)(36 46)(37 43)(38 44)(39 41)(40 42)(49 63)(50 64)(51 61)(52 62)(53 59)(54 60)(55 57)(56 58)(65 79)(66 80)(67 77)(68 78)(69 75)(70 76)(71 73)(72 74)(81 95)(82 96)(83 93)(84 94)(85 91)(86 92)(87 89)(88 90)(97 111)(98 112)(99 109)(100 110)(101 107)(102 108)(103 105)(104 106)(113 127)(114 128)(115 125)(116 126)(117 123)(118 124)(119 121)(120 122)
(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 27 3 25)(2 26 4 28)(5 29 7 31)(6 32 8 30)(9 17 11 19)(10 20 12 18)(13 21 15 23)(14 24 16 22)(33 57 35 59)(34 60 36 58)(37 61 39 63)(38 64 40 62)(41 49 43 51)(42 52 44 50)(45 53 47 55)(46 56 48 54)(65 92 67 90)(66 91 68 89)(69 96 71 94)(70 95 72 93)(73 84 75 82)(74 83 76 81)(77 88 79 86)(78 87 80 85)(97 124 99 122)(98 123 100 121)(101 128 103 126)(102 127 104 125)(105 116 107 114)(106 115 108 113)(109 120 111 118)(110 119 112 117)
(1 45 5 43)(2 46 6 44)(3 47 7 41)(4 48 8 42)(9 39 15 33)(10 40 16 34)(11 37 13 35)(12 38 14 36)(17 63 23 57)(18 64 24 58)(19 61 21 59)(20 62 22 60)(25 55 31 49)(26 56 32 50)(27 53 29 51)(28 54 30 52)(65 107 71 109)(66 108 72 110)(67 105 69 111)(68 106 70 112)(73 99 79 101)(74 100 80 102)(75 97 77 103)(76 98 78 104)(81 123 87 125)(82 124 88 126)(83 121 85 127)(84 122 86 128)(89 115 95 117)(90 116 96 118)(91 113 93 119)(92 114 94 120)
(1 77 7 73)(2 80 8 76)(3 79 5 75)(4 78 6 74)(9 71 13 67)(10 70 14 66)(11 69 15 65)(12 72 16 68)(17 95 21 91)(18 94 22 90)(19 93 23 89)(20 96 24 92)(25 87 29 83)(26 86 30 82)(27 85 31 81)(28 88 32 84)(33 111 37 107)(34 110 38 106)(35 109 39 105)(36 112 40 108)(41 103 45 99)(42 102 46 98)(43 101 47 97)(44 104 48 100)(49 127 53 123)(50 126 54 122)(51 125 55 121)(52 128 56 124)(57 119 61 115)(58 118 62 114)(59 117 63 113)(60 120 64 116)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111222224
type++++++++++--
imageC1C2C2C2C2C2C2C2D4D4Q8C4○D4C4○D8C8.C22
kernelC2×Q8.Q8C2×Q8⋊C4C2×C4⋊C8C2×C4.Q8C2×C2.D8Q8.Q8C2×C4×Q8C2×C42.C2C42C22×C4C2×Q8C2×C4C22C22
# reps12111811224482

Matrix representation of C2×Q8.Q8 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
001000
000100
000001
0000160
,
0160000
1600000
001000
000100
000040
0000013
,
100000
010000
0041500
0001300
0000130
0000013
,
1450000
1230000
0012300
0014500
0000143
000033

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,15,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,12,0,0,0,0,5,3,0,0,0,0,0,0,12,14,0,0,0,0,3,5,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;

C2×Q8.Q8 in GAP, Magma, Sage, TeX

C_2\times Q_8.Q_8
% in TeX

G:=Group("C2xQ8.Q8");
// GroupNames label

G:=SmallGroup(128,1807);
// by ID

G=gap.SmallGroup(128,1807);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,352,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽