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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×Q8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4×Q8 — C2×Q8.Q8
 Lower central C1 — C2 — C2×C4 — C2×Q8.Q8
 Upper central C1 — C23 — C2×C42 — C2×Q8.Q8
 Jennings C1 — C2 — C2 — C2×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 ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 Q8 C4○D4 C4○D8 C8.C22 kernel C2×Q8.Q8 C2×Q8⋊C4 C2×C4⋊C8 C2×C4.Q8 C2×C2.D8 Q8.Q8 C2×C4×Q8 C2×C42.C2 C42 C22×C4 C2×Q8 C2×C4 C22 C22 # reps 1 2 1 1 1 8 1 1 2 2 4 4 8 2

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

 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 1 0 0 0 0 16 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 15 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 14 5 0 0 0 0 12 3 0 0 0 0 0 0 12 3 0 0 0 0 14 5 0 0 0 0 0 0 14 3 0 0 0 0 3 3

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

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