Copied to
clipboard

G = (C2×C8).Q8order 128 = 27

30th non-split extension by C2×C8 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for (C2×C8).Q8
G = < a,b,c,d | a2=b8=1, c4=b4, d2=b4c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=ab5, dcd-1=b6c3 >

Subgroups: 180 in 114 conjugacy classes, 60 normal (52 characteristic)
C1, C2 [×7], C4 [×4], C4 [×8], C22 [×7], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×16], C23, C42 [×6], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×10], C22×C4 [×7], C2.C42 [×2], C8⋊C4 [×2], C4⋊C8 [×2], C2×C42 [×3], C2×C4⋊C4 [×2], C22×C8 [×4], C22.7C42 [×4], C4×C4⋊C4, C2×C8⋊C4, C2×C4⋊C8, (C2×C8).Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, M4(2) [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C2×M4(2) [×2], C8○D4 [×2], C23.63C23, C42.6C4, C42.7C22, C89D4, C86D4, C84Q8 [×2], (C2×C8).Q8

Smallest permutation representation of (C2×C8).Q8
Regular action on 128 points
Generators in S128
(1 100)(2 101)(3 102)(4 103)(5 104)(6 97)(7 98)(8 99)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(25 95)(26 96)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 119)(50 120)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 73)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 121)
(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 34 31 19 5 38 27 23)(2 39 32 24 6 35 28 20)(3 36 25 21 7 40 29 17)(4 33 26 18 8 37 30 22)(9 113 59 66 13 117 63 70)(10 118 60 71 14 114 64 67)(11 115 61 68 15 119 57 72)(12 120 62 65 16 116 58 69)(41 96 84 99 45 92 88 103)(42 93 85 104 46 89 81 100)(43 90 86 101 47 94 82 97)(44 95 87 98 48 91 83 102)(49 74 121 107 53 78 125 111)(50 79 122 112 54 75 126 108)(51 76 123 109 55 80 127 105)(52 73 124 106 56 77 128 110)
(1 60 27 10 5 64 31 14)(2 74 28 111 6 78 32 107)(3 62 29 12 7 58 25 16)(4 76 30 105 8 80 26 109)(9 99 63 96 13 103 59 92)(11 101 57 90 15 97 61 94)(17 71 40 118 21 67 36 114)(18 125 33 53 22 121 37 49)(19 65 34 120 23 69 38 116)(20 127 35 55 24 123 39 51)(41 115 88 72 45 119 84 68)(42 50 81 126 46 54 85 122)(43 117 82 66 47 113 86 70)(44 52 83 128 48 56 87 124)(73 93 110 100 77 89 106 104)(75 95 112 102 79 91 108 98)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4T 8A ··· 8P order 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 M4(2) C4○D4 C8○D4 kernel (C2×C8).Q8 C22.7C42 C4×C4⋊C4 C2×C8⋊C4 C2×C4⋊C8 C2.C42 C2×C4⋊C4 C2×C8 C2×C8 C2×C4 C2×C4 C22 # reps 1 4 1 1 1 4 4 2 2 8 8 8

Matrix representation of (C2×C8).Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 5 8 0 0 0 0 8 12 0 0 0 0 0 0 13 9 0 0 0 0 4 4
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 2 3 0 0 0 0 3 15 0 0 0 0 0 0 1 2 0 0 0 0 16 16
,
 6 1 0 0 0 0 16 11 0 0 0 0 0 0 12 9 0 0 0 0 9 5 0 0 0 0 0 0 13 9 0 0 0 0 0 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,5,8,0,0,0,0,8,12,0,0,0,0,0,0,13,4,0,0,0,0,9,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,3,0,0,0,0,3,15,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[6,16,0,0,0,0,1,11,0,0,0,0,0,0,12,9,0,0,0,0,9,5,0,0,0,0,0,0,13,0,0,0,0,0,9,4] >;

(C2×C8).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_8).Q_8
% in TeX

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

G:=SmallGroup(128,649);
// by ID

G=gap.SmallGroup(128,649);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,723,58,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=b^4*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=a*b^5,d*c*d^-1=b^6*c^3>;
// generators/relations

׿
×
𝔽