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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, C42.216D4, C42.329C23, Q81(C2×Q8), (C2×Q8)⋊13Q8, C4⋊C4.36C23, C4.84(C2×SD16), C4.24(C22×Q8), C4⋊C8.333C22, (C2×C4).271C24, (C2×C8).308C23, (C2×C4).122SD16, C23.867(C2×D4), (C22×C4).797D4, C4⋊Q8.258C22, C4.64(C22⋊Q8), (C4×Q8).293C22, (C2×Q8).361C23, C2.11(C22×SD16), C22.85(C2×SD16), C4.Q8.145C22, (C22×C8).343C22, (C2×C42).817C22, C22.531(C22×D4), C22.99(C22⋊Q8), (C22×C4).1541C23, Q8⋊C4.171C22, (C22×Q8).471C22, C22.107(C8.C22), (C2×C4⋊C8).50C2, (C2×C4×Q8).49C2, C4.81(C2×C4○D4), (C2×C4⋊Q8).43C2, (C2×C4).319(C2×Q8), (C2×C4.Q8).31C2, C2.52(C2×C22⋊Q8), (C2×C4).1433(C2×D4), C2.21(C2×C8.C22), (C2×C4).837(C4○D4), (C2×C4⋊C4).600C22, (C2×Q8⋊C4).29C2, SmallGroup(128,1805)

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=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >

Subgroups: 348 in 208 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×20], Q8 [×4], Q8 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Q8 [×11], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×8], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8 [×2], C22×Q8, C22×Q8, C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4.Q8 [×2], Q8⋊Q8 [×8], C2×C4×Q8, C2×C4⋊Q8, C2×Q8⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C2×SD16 [×6], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, Q8⋊Q8 [×4], C2×C22⋊Q8, C22×SD16, 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 47 7 43)(2 48 8 44)(3 45 5 41)(4 46 6 42)(9 37 13 33)(10 38 14 34)(11 39 15 35)(12 40 16 36)(17 61 21 57)(18 62 22 58)(19 63 23 59)(20 64 24 60)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)(65 105 69 109)(66 106 70 110)(67 107 71 111)(68 108 72 112)(73 97 77 101)(74 98 78 102)(75 99 79 103)(76 100 80 104)(81 121 85 125)(82 122 86 126)(83 123 87 127)(84 124 88 128)(89 113 93 117)(90 114 94 118)(91 115 95 119)(92 116 96 120)
(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), (1,47,7,43)(2,48,8,44)(3,45,5,41)(4,46,6,42)(9,37,13,33)(10,38,14,34)(11,39,15,35)(12,40,16,36)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(65,105,69,109)(66,106,70,110)(67,107,71,111)(68,108,72,112)(73,97,77,101)(74,98,78,102)(75,99,79,103)(76,100,80,104)(81,121,85,125)(82,122,86,126)(83,123,87,127)(84,124,88,128)(89,113,93,117)(90,114,94,118)(91,115,95,119)(92,116,96,120), (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), (1,47,7,43)(2,48,8,44)(3,45,5,41)(4,46,6,42)(9,37,13,33)(10,38,14,34)(11,39,15,35)(12,40,16,36)(17,61,21,57)(18,62,22,58)(19,63,23,59)(20,64,24,60)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(65,105,69,109)(66,106,70,110)(67,107,71,111)(68,108,72,112)(73,97,77,101)(74,98,78,102)(75,99,79,103)(76,100,80,104)(81,121,85,125)(82,122,86,126)(83,123,87,127)(84,124,88,128)(89,113,93,117)(90,114,94,118)(91,115,95,119)(92,116,96,120), (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)], [(1,47,7,43),(2,48,8,44),(3,45,5,41),(4,46,6,42),(9,37,13,33),(10,38,14,34),(11,39,15,35),(12,40,16,36),(17,61,21,57),(18,62,22,58),(19,63,23,59),(20,64,24,60),(25,53,29,49),(26,54,30,50),(27,55,31,51),(28,56,32,52),(65,105,69,109),(66,106,70,110),(67,107,71,111),(68,108,72,112),(73,97,77,101),(74,98,78,102),(75,99,79,103),(76,100,80,104),(81,121,85,125),(82,122,86,126),(83,123,87,127),(84,124,88,128),(89,113,93,117),(90,114,94,118),(91,115,95,119),(92,116,96,120)], [(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

dim1111111222224
type+++++++++--
imageC1C2C2C2C2C2C2D4D4Q8SD16C4○D4C8.C22
kernelC2×Q8⋊Q8C2×Q8⋊C4C2×C4⋊C8C2×C4.Q8Q8⋊Q8C2×C4×Q8C2×C4⋊Q8C42C22×C4C2×Q8C2×C4C2×C4C22
# reps1212811224842

Matrix representation of C2×Q8⋊Q8 in GL5(𝔽17)

160000
01000
00100
000160
000016
,
10000
01000
00100
000115
000116
,
10000
016000
001600
000815
00079
,
160000
001600
01000
000160
000016
,
10000
07100
011000
000109
00067

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,15,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,8,7,0,0,0,15,9],[16,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,7,1,0,0,0,1,10,0,0,0,0,0,10,6,0,0,0,9,7] >;

C2×Q8⋊Q8 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes Q_8
% in TeX

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

G:=SmallGroup(128,1805);
// by ID

G=gap.SmallGroup(128,1805);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,352,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽