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## G = C7×C4.10C42order 448 = 26·7

### Direct product of C7 and C4.10C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C4.10C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C28 — C14×M4(2) — C7×C4.10C42
 Lower central C1 — C4 — C7×C4.10C42
 Upper central C1 — C28 — C7×C4.10C42

Generators and relations for C7×C4.10C42
G = < a,b,c,d | a7=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×M4(2), C56, C2×C28, C22×C14, C4.10C42, C2×C56, C7×M4(2), C22×C28, C14×M4(2), C7×C4.10C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C2×C28, C7×D4, C7×Q8, C4.10C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C4.10C42

Smallest permutation representation of C7×C4.10C42
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 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 79 77 75)(74 80 78 76)(81 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(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 97 3 103 5 101 7 99)(2 104 8 98 6 100 4 102)(9 44 11 42 13 48 15 46)(10 47 16 41 14 43 12 45)(17 55 19 53 21 51 23 49)(18 54 24 56 22 50 20 52)(25 59 27 57 29 63 31 61)(26 58 32 60 30 62 28 64)(33 87 35 85 37 83 39 81)(34 86 40 88 38 82 36 84)(65 89 71 91 69 93 67 95)(66 92 68 90 70 96 72 94)(73 111 75 109 77 107 79 105)(74 110 80 112 78 106 76 108)

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

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

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

154 conjugacy classes

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

154 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + - image C1 C2 C4 C7 C14 C28 D4 Q8 C7×D4 C7×Q8 C4.10C42 C7×C4.10C42 kernel C7×C4.10C42 C14×M4(2) C2×C56 C4.10C42 C2×M4(2) C2×C8 C2×C28 C22×C14 C2×C4 C23 C7 C1 # reps 1 3 12 6 18 72 3 1 18 6 2 12

Matrix representation of C7×C4.10C42 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 112 0 111 0 0 0 1 1 8 0 1 0 7 15 112 0
,
 112 111 0 0 8 1 0 0 106 1 0 15 0 112 1 0
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[112,0,8,7,0,0,0,15,111,1,1,112,0,1,0,0],[112,8,106,0,111,1,1,112,0,0,0,1,0,0,15,0] >;

C7×C4.10C42 in GAP, Magma, Sage, TeX

C_7\times C_4._{10}C_4^2
% in TeX

G:=Group("C7xC4.10C4^2");
// GroupNames label

G:=SmallGroup(448,142);
// by ID

G=gap.SmallGroup(448,142);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,248,4911,172,14117,124]);
// Polycyclic

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

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