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## G = C2×C8⋊2Q8order 128 = 27

### Direct product of C2 and C8⋊2Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C4×C8 — C2×C8⋊2Q8
 Lower central C1 — C2 — C2×C4 — C2×C8⋊2Q8
 Upper central C1 — C23 — C2×C42 — C2×C8⋊2Q8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊2Q8

Generators and relations for C2×C82Q8
G = < a,b,c,d | a2=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 372 in 212 conjugacy classes, 132 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C2×C4 [×16], Q8 [×16], C23, C42 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×16], C4×C8 [×4], C2.D8 [×16], C2×C42, C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4⋊Q8 [×8], C4⋊Q8 [×4], C22×C8 [×2], C22×Q8 [×2], C2×C4×C8, C2×C2.D8 [×4], C82Q8 [×8], C2×C4⋊Q8 [×2], C2×C82Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D8 [×4], Q16 [×4], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C2×D8 [×6], C2×Q16 [×6], C22×D4, C22×Q8 [×2], C82Q8 [×4], C2×C4⋊Q8, C22×D8, C22×Q16, C2×C82Q8

Smallest permutation representation of C2×C82Q8
Regular action on 128 points
Generators in S128
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 97)(32 98)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 105)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 121)
(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 59 43 34)(2 60 44 35)(3 61 45 36)(4 62 46 37)(5 63 47 38)(6 64 48 39)(7 57 41 40)(8 58 42 33)(9 80 18 53)(10 73 19 54)(11 74 20 55)(12 75 21 56)(13 76 22 49)(14 77 23 50)(15 78 24 51)(16 79 17 52)(25 71 113 94)(26 72 114 95)(27 65 115 96)(28 66 116 89)(29 67 117 90)(30 68 118 91)(31 69 119 92)(32 70 120 93)(81 103 108 126)(82 104 109 127)(83 97 110 128)(84 98 111 121)(85 99 112 122)(86 100 105 123)(87 101 106 124)(88 102 107 125)
(1 93 43 70)(2 92 44 69)(3 91 45 68)(4 90 46 67)(5 89 47 66)(6 96 48 65)(7 95 41 72)(8 94 42 71)(9 108 18 81)(10 107 19 88)(11 106 20 87)(12 105 21 86)(13 112 22 85)(14 111 23 84)(15 110 24 83)(16 109 17 82)(25 58 113 33)(26 57 114 40)(27 64 115 39)(28 63 116 38)(29 62 117 37)(30 61 118 36)(31 60 119 35)(32 59 120 34)(49 122 76 99)(50 121 77 98)(51 128 78 97)(52 127 79 104)(53 126 80 103)(54 125 73 102)(55 124 74 101)(56 123 75 100)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 D4 Q8 D4 D8 Q16 kernel C2×C8⋊2Q8 C2×C4×C8 C2×C2.D8 C8⋊2Q8 C2×C4⋊Q8 C42 C2×C8 C22×C4 C2×C4 C2×C4 # reps 1 1 4 8 2 2 8 2 8 8

Matrix representation of C2×C82Q8 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 15 0 0 0 1 1 0 0 0 0 0 3 14 0 0 0 3 3
,
 1 0 0 0 0 0 16 15 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 9 3 0 0 0 1 8 0 0 0 0 0 5 5 0 0 0 5 12

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,16,1,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,9,1,0,0,0,3,8,0,0,0,0,0,5,5,0,0,0,5,12] >;

C2×C82Q8 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2Q_8
% in TeX

G:=Group("C2xC8:2Q8");
// GroupNames label

G:=SmallGroup(128,1891);
// by ID

G=gap.SmallGroup(128,1891);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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