Copied to
clipboard

## G = C4×C2.D8order 128 = 27

### Direct product of C4 and C2.D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C4×C2.D8
 Lower central C1 — C2 — C4 — C4×C2.D8
 Upper central C1 — C22×C4 — C2×C42 — C4×C2.D8
 Jennings C1 — C2 — C2 — C22×C4 — C4×C2.D8

Generators and relations for C4×C2.D8
G = < a,b,c,d | a4=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 140 conjugacy classes, 92 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×18], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×4], C2.D8 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C22.4Q16 [×2], C4×C4⋊C4 [×2], C2×C4×C8, C2×C2.D8 [×2], C4×C2.D8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2.D8 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×D8, C2×Q16, C4○D8 [×2], C4×C4⋊C4, C2×C2.D8, C23.25D4, C4×D8 [×2], C4×Q16 [×2], C4×C2.D8

Smallest permutation representation of C4×C2.D8
Regular action on 128 points
Generators in S128
(1 21 47 111)(2 22 48 112)(3 23 41 105)(4 24 42 106)(5 17 43 107)(6 18 44 108)(7 19 45 109)(8 20 46 110)(9 85 62 99)(10 86 63 100)(11 87 64 101)(12 88 57 102)(13 81 58 103)(14 82 59 104)(15 83 60 97)(16 84 61 98)(25 67 113 33)(26 68 114 34)(27 69 115 35)(28 70 116 36)(29 71 117 37)(30 72 118 38)(31 65 119 39)(32 66 120 40)(49 89 123 76)(50 90 124 77)(51 91 125 78)(52 92 126 79)(53 93 127 80)(54 94 128 73)(55 95 121 74)(56 96 122 75)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 77)(10 78)(11 79)(12 80)(13 73)(14 74)(15 75)(16 76)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(49 84)(50 85)(51 86)(52 87)(53 88)(54 81)(55 82)(56 83)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 105)(72 106)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 121)
(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 97 115 122)(2 104 116 121)(3 103 117 128)(4 102 118 127)(5 101 119 126)(6 100 120 125)(7 99 113 124)(8 98 114 123)(9 33 77 19)(10 40 78 18)(11 39 79 17)(12 38 80 24)(13 37 73 23)(14 36 74 22)(15 35 75 21)(16 34 76 20)(25 50 45 85)(26 49 46 84)(27 56 47 83)(28 55 48 82)(29 54 41 81)(30 53 42 88)(31 52 43 87)(32 51 44 86)(57 72 93 106)(58 71 94 105)(59 70 95 112)(60 69 96 111)(61 68 89 110)(62 67 90 109)(63 66 91 108)(64 65 92 107)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 Q8 D4 D8 Q16 C4○D4 C4○D8 kernel C4×C2.D8 C22.4Q16 C4×C4⋊C4 C2×C4×C8 C2×C2.D8 C4×C8 C2.D8 C42 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 2 1 2 8 16 2 2 4 4 4 8

Matrix representation of C4×C2.D8 in GL5(𝔽17)

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

G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,12,12,0,0,0,12,5,0,0,0,0,0,3,3,0,0,0,14,3],[1,0,0,0,0,0,5,12,0,0,0,12,12,0,0,0,0,0,0,13,0,0,0,13,0] >;

C4×C2.D8 in GAP, Magma, Sage, TeX

C_4\times C_2.D_8
% in TeX

G:=Group("C4xC2.D8");
// GroupNames label

G:=SmallGroup(128,507);
// by ID

G=gap.SmallGroup(128,507);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,478,1018,248,1411,172]);
// Polycyclic

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

׿
×
𝔽