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## G = C7×C4.10D4order 224 = 25·7

### Direct product of C7 and C4.10D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4.10D4
 Lower central C1 — C2 — C22 — C7×C4.10D4
 Upper central C1 — C14 — C2×C28 — C7×C4.10D4

Generators and relations for C7×C4.10D4
G = < a,b,c,d | a7=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C7×C4.10D4
On 112 points
Generators in S112
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 106 66 50 82 98 62)(10 107 67 51 83 99 63)(11 108 68 52 84 100 64)(12 109 69 53 85 101 57)(13 110 70 54 86 102 58)(14 111 71 55 87 103 59)(15 112 72 56 88 104 60)(16 105 65 49 81 97 61)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(74 80 78 76)(81 83 85 87)(82 88 86 84)(89 91 93 95)(90 96 94 92)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)
(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)
(1 100 7 102 5 104 3 98)(2 101 4 99 6 97 8 103)(9 47 11 45 13 43 15 41)(10 44 16 46 14 48 12 42)(17 50 23 52 21 54 19 56)(18 51 20 49 22 55 24 53)(25 62 31 64 29 58 27 60)(26 63 28 61 30 59 32 57)(33 82 39 84 37 86 35 88)(34 83 36 81 38 87 40 85)(65 94 71 96 69 90 67 92)(66 95 68 93 70 91 72 89)(73 106 79 108 77 110 75 112)(74 107 76 105 78 111 80 109)

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

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

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

C7×C4.10D4 is a maximal subgroup of   (C2×C4).D28  (C2×Q8).D14  M4(2).21D14  D28.4D4  D28.5D4  D28.6D4  D28.7D4

77 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 14A ··· 14F 14G ··· 14L 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 4 4 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + - image C1 C2 C2 C4 C7 C14 C14 C28 D4 C7×D4 C4.10D4 C7×C4.10D4 kernel C7×C4.10D4 C7×M4(2) Q8×C14 C2×C28 C4.10D4 M4(2) C2×Q8 C2×C4 C28 C4 C7 C1 # reps 1 2 1 4 6 12 6 24 2 12 1 6

Matrix representation of C7×C4.10D4 in GL4(𝔽113) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 0 1 0 0 112 0 0 0 108 103 0 112 10 108 1 0
,
 108 10 111 0 0 98 0 111 69 69 5 103 57 56 0 15
,
 55 6 22 23 1 59 23 91 3 3 60 2 111 100 110 52
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[0,112,108,10,1,0,103,108,0,0,0,1,0,0,112,0],[108,0,69,57,10,98,69,56,111,0,5,0,0,111,103,15],[55,1,3,111,6,59,3,100,22,23,60,110,23,91,2,52] >;

C7×C4.10D4 in GAP, Magma, Sage, TeX

C_7\times C_4._{10}D_4
% in TeX

G:=Group("C7xC4.10D4");
// GroupNames label

G:=SmallGroup(224,50);
// by ID

G=gap.SmallGroup(224,50);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,679,3363,2530,88]);
// Polycyclic

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

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