Copied to
clipboard

## G = C7×C8○D8order 448 = 26·7

### Direct product of C7 and C8○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C8○D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4≀C2 — C7×C8○D8
 Lower central C1 — C2 — C4 — C7×C8○D8
 Upper central C1 — C56 — C2×C56 — C7×C8○D8

Generators and relations for C7×C8○D8
G = < a,b,c,d | a7=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >

Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8○D8, C4×C28, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C4×C56, C7×C4≀C2, C7×C8.C4, C7×C8○D4, C7×C4○D8, C7×C8○D8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×C28, C7×D4, C22×C14, C8○D8, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8○D8

Smallest permutation representation of C7×C8○D8
On 112 points
Generators in S112
(1 59 23 55 10 47 35)(2 60 24 56 11 48 36)(3 61 17 49 12 41 37)(4 62 18 50 13 42 38)(5 63 19 51 14 43 39)(6 64 20 52 15 44 40)(7 57 21 53 16 45 33)(8 58 22 54 9 46 34)(25 76 105 96 85 68 97)(26 77 106 89 86 69 98)(27 78 107 90 87 70 99)(28 79 108 91 88 71 100)(29 80 109 92 81 72 101)(30 73 110 93 82 65 102)(31 74 111 94 83 66 103)(32 75 112 95 84 67 104)
(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)
(1 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12)(17 22 19 24 21 18 23 20)(25 28 31 26 29 32 27 30)(33 38 35 40 37 34 39 36)(41 46 43 48 45 42 47 44)(49 54 51 56 53 50 55 52)(57 62 59 64 61 58 63 60)(65 68 71 66 69 72 67 70)(73 76 79 74 77 80 75 78)(81 84 87 82 85 88 83 86)(89 92 95 90 93 96 91 94)(97 100 103 98 101 104 99 102)(105 108 111 106 109 112 107 110)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)

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

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

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

196 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C ··· 4G 4H 4I 7A ··· 7F 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 14A ··· 14F 14G ··· 14L 14M ··· 14X 28A ··· 28L 28M ··· 28AP 28AQ ··· 28BB 56A ··· 56X 56Y ··· 56BH 56BI ··· 56CF order 1 2 2 2 2 4 4 4 ··· 4 4 4 7 ··· 7 8 8 8 8 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 56 ··· 56 size 1 1 2 4 4 1 1 2 ··· 2 4 4 1 ··· 1 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C14 C14 C28 C28 C28 D4 C4○D4 C7×D4 C8○D8 C7×C4○D4 C7×C8○D8 kernel C7×C8○D8 C4×C56 C7×C4≀C2 C7×C8.C4 C7×C8○D4 C7×C4○D8 C7×D8 C7×SD16 C7×Q16 C8○D8 C4×C8 C4≀C2 C8.C4 C8○D4 C4○D8 D8 SD16 Q16 C56 C2×C14 C8 C7 C22 C1 # reps 1 1 2 1 2 1 2 4 2 6 6 12 6 12 6 12 24 12 2 2 12 8 12 48

Matrix representation of C7×C8○D8 in GL3(𝔽113) generated by

 30 0 0 0 1 0 0 0 1
,
 1 0 0 0 18 0 0 0 18
,
 112 0 0 0 95 0 0 13 69
,
 112 0 0 0 1 111 0 0 112
G:=sub<GL(3,GF(113))| [30,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,18],[112,0,0,0,95,13,0,0,69],[112,0,0,0,1,0,0,111,112] >;

C7×C8○D8 in GAP, Magma, Sage, TeX

C_7\times C_8\circ D_8
% in TeX

G:=Group("C7xC8oD8");
// GroupNames label

G:=SmallGroup(448,851);
// by ID

G=gap.SmallGroup(448,851);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,604,9804,4911,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations

׿
×
𝔽