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## G = C2×Q82order 128 = 27

### Direct product of C2, Q8 and Q8

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×Q82
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×Q8 — C2×Q82
 Lower central C1 — C22 — C2×Q82
 Upper central C1 — C23 — C2×Q82
 Jennings C1 — C22 — C2×Q82

Generators and relations for C2×Q82
G = < a,b,c,d,e | a2=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 700 in 556 conjugacy classes, 484 normal (5 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C22×C4, C2×Q8, C2×Q8, C2×C42, C2×C4⋊C4, C4×Q8, C4⋊Q8, C22×Q8, C2×C4×Q8, C2×C4⋊Q8, Q82, C2×Q82
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, C25, Q82, Q8×C23, C2×2+ 1+4, C2×Q82

Smallest permutation representation of C2×Q82
Regular action on 128 points
Generators in S128
(1 17)(2 18)(3 19)(4 20)(5 26)(6 27)(7 28)(8 25)(9 37)(10 38)(11 39)(12 40)(13 32)(14 29)(15 30)(16 31)(21 109)(22 110)(23 111)(24 112)(33 42)(34 43)(35 44)(36 41)(45 71)(46 72)(47 69)(48 70)(49 58)(50 59)(51 60)(52 57)(53 62)(54 63)(55 64)(56 61)(65 76)(66 73)(67 74)(68 75)(77 104)(78 101)(79 102)(80 103)(81 92)(82 89)(83 90)(84 91)(85 96)(86 93)(87 94)(88 95)(97 108)(98 105)(99 106)(100 107)(113 124)(114 121)(115 122)(116 123)(117 128)(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 86 3 88)(2 85 4 87)(5 102 7 104)(6 101 8 103)(9 67 11 65)(10 66 12 68)(13 81 15 83)(14 84 16 82)(17 93 19 95)(18 96 20 94)(21 41 23 43)(22 44 24 42)(25 80 27 78)(26 79 28 77)(29 91 31 89)(30 90 32 92)(33 110 35 112)(34 109 36 111)(37 74 39 76)(38 73 40 75)(45 121 47 123)(46 124 48 122)(49 119 51 117)(50 118 52 120)(53 106 55 108)(54 105 56 107)(57 127 59 125)(58 126 60 128)(61 100 63 98)(62 99 64 97)(69 116 71 114)(70 115 72 113)
(1 12 6 29)(2 9 7 30)(3 10 8 31)(4 11 5 32)(13 20 39 26)(14 17 40 27)(15 18 37 28)(16 19 38 25)(21 99 117 122)(22 100 118 123)(23 97 119 124)(24 98 120 121)(33 56 59 69)(34 53 60 70)(35 54 57 71)(36 55 58 72)(41 64 49 46)(42 61 50 47)(43 62 51 48)(44 63 52 45)(65 102 92 87)(66 103 89 88)(67 104 90 85)(68 101 91 86)(73 80 82 95)(74 77 83 96)(75 78 84 93)(76 79 81 94)(105 127 114 112)(106 128 115 109)(107 125 116 110)(108 126 113 111)
(1 35 6 57)(2 36 7 58)(3 33 8 59)(4 34 5 60)(9 72 30 55)(10 69 31 56)(11 70 32 53)(12 71 29 54)(13 62 39 48)(14 63 40 45)(15 64 37 46)(16 61 38 47)(17 44 27 52)(18 41 28 49)(19 42 25 50)(20 43 26 51)(21 79 117 94)(22 80 118 95)(23 77 119 96)(24 78 120 93)(65 115 92 106)(66 116 89 107)(67 113 90 108)(68 114 91 105)(73 123 82 100)(74 124 83 97)(75 121 84 98)(76 122 81 99)(85 111 104 126)(86 112 101 127)(87 109 102 128)(88 110 103 125)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 4Y ··· 4AP order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 2 4 type + + + + - + image C1 C2 C2 C2 Q8 2+ 1+4 kernel C2×Q82 C2×C4×Q8 C2×C4⋊Q8 Q82 C2×Q8 C22 # reps 1 6 9 16 16 2

Matrix representation of C2×Q82 in GL6(𝔽5)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 2
,
 3 0 0 0 0 0 2 2 0 0 0 0 0 0 2 0 0 0 0 0 1 3 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 2 4 0 0 0 0 0 3 0 0 0 0 0 0 2 2 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 4

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C2×Q82 in GAP, Magma, Sage, TeX

C_2\times Q_8^2
% in TeX

G:=Group("C2xQ8^2");
// GroupNames label

G:=SmallGroup(128,2209);
// by ID

G=gap.SmallGroup(128,2209);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,672,477,232,1430,352,570,136]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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