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## G = C7×C22⋊C8order 224 = 25·7

### Direct product of C7 and C22⋊C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C22⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C2×C56 — C7×C22⋊C8
 Lower central C1 — C2 — C7×C22⋊C8
 Upper central C1 — C2×C28 — C7×C22⋊C8

Generators and relations for C7×C22⋊C8
G = < a,b,c,d | a7=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C7×C22⋊C8
On 112 points
Generators in S112
(1 69 16 81 25 73 17)(2 70 9 82 26 74 18)(3 71 10 83 27 75 19)(4 72 11 84 28 76 20)(5 65 12 85 29 77 21)(6 66 13 86 30 78 22)(7 67 14 87 31 79 23)(8 68 15 88 32 80 24)(33 61 105 49 97 41 89)(34 62 106 50 98 42 90)(35 63 107 51 99 43 91)(36 64 108 52 100 44 92)(37 57 109 53 101 45 93)(38 58 110 54 102 46 94)(39 59 111 55 103 47 95)(40 60 112 56 104 48 96)
(2 40)(4 34)(6 36)(8 38)(9 112)(11 106)(13 108)(15 110)(18 96)(20 90)(22 92)(24 94)(26 104)(28 98)(30 100)(32 102)(42 76)(44 78)(46 80)(48 74)(50 84)(52 86)(54 88)(56 82)(58 68)(60 70)(62 72)(64 66)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 112)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 95)(18 96)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 103)(26 104)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 65)(64 66)
(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)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C4 C4 C7 C8 C14 C14 C28 C28 C56 D4 M4(2) C7×D4 C7×M4(2) kernel C7×C22⋊C8 C2×C56 C22×C28 C2×C28 C22×C14 C22⋊C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 C28 C14 C4 C2 # reps 1 2 1 2 2 6 8 12 6 12 12 48 2 2 12 12

Matrix representation of C7×C22⋊C8 in GL4(𝔽113) generated by

 30 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 52 112
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 44 0 0 0 0 52 111 0 0 53 61
G:=sub<GL(4,GF(113))| [30,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,52,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,44,0,0,0,0,52,53,0,0,111,61] >;

C7×C22⋊C8 in GAP, Magma, Sage, TeX

C_7\times C_2^2\rtimes C_8
% in TeX

G:=Group("C7xC2^2:C8");
// GroupNames label

G:=SmallGroup(224,47);
// by ID

G=gap.SmallGroup(224,47);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,88]);
// Polycyclic

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

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