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## G = C7×C4.9C42order 448 = 26·7

### Direct product of C7 and C4.9C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C4.9C42
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C28 — C7×C42⋊C2 — C7×C4.9C42
 Lower central C1 — C4 — C7×C4.9C42
 Upper central C1 — C28 — C7×C4.9C42

Generators and relations for C7×C4.9C42
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 154 in 94 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C56, C2×C28, C2×C28, C2×C28, C22×C14, C4.9C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, C7×C42⋊C2, C14×M4(2), C7×C4.9C42
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, C4.9C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C4.9C42

Smallest permutation representation of C7×C4.9C42
On 112 points
Generators in S112
(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 50 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 100 25 44)(9 101 26 45)(10 102 27 46)(11 103 28 47)(12 104 22 48)(13 105 23 49)(14 99 24 43)(15 35 109 42)(16 29 110 36)(17 30 111 37)(18 31 112 38)(19 32 106 39)(20 33 107 40)(21 34 108 41)(64 90 71 98)(65 91 72 92)(66 85 73 93)(67 86 74 94)(68 87 75 95)(69 88 76 96)(70 89 77 97)
(1 22 68 32)(2 23 69 33)(3 24 70 34)(4 25 64 35)(5 26 65 29)(6 27 66 30)(7 28 67 31)(8 71 42 81)(9 72 36 82)(10 73 37 83)(11 74 38 84)(12 75 39 78)(13 76 40 79)(14 77 41 80)(15 62 100 98)(16 63 101 92)(17 57 102 93)(18 58 103 94)(19 59 104 95)(20 60 105 96)(21 61 99 97)(43 89 108 52)(44 90 109 53)(45 91 110 54)(46 85 111 55)(47 86 112 56)(48 87 106 50)(49 88 107 51)
(8 44 25 100)(9 45 26 101)(10 46 27 102)(11 47 28 103)(12 48 22 104)(13 49 23 105)(14 43 24 99)(15 35 109 42)(16 29 110 36)(17 30 111 37)(18 31 112 38)(19 32 106 39)(20 33 107 40)(21 34 108 41)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)

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

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

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

154 conjugacy classes

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

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C4 C4 C7 C14 C14 C28 C28 D4 Q8 D4 C7×D4 C7×Q8 C7×D4 C4.9C42 C7×C4.9C42 kernel C7×C4.9C42 C7×C42⋊C2 C14×M4(2) C4×C28 C2×C56 C4.9C42 C42⋊C2 C2×M4(2) C42 C2×C8 C2×C28 C2×C28 C22×C14 C2×C4 C2×C4 C23 C7 C1 # reps 1 2 1 8 4 6 12 6 48 24 2 1 1 12 6 6 2 12

Matrix representation of C7×C4.9C42 in GL4(𝔽113) generated by

 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 0 0 1 0 0 0 0 1 112 111 0 0 0 1 0 0
,
 1 0 0 0 112 112 0 0 0 0 98 0 0 0 15 15
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,112,0,0,0,111,1,1,0,0,0,0,1,0,0],[1,112,0,0,0,112,0,0,0,0,98,15,0,0,0,15] >;

C7×C4.9C42 in GAP, Magma, Sage, TeX

C_7\times C_4._9C_4^2
% in TeX

G:=Group("C7xC4.9C4^2");
// GroupNames label

G:=SmallGroup(448,141);
// by ID

G=gap.SmallGroup(448,141);
# by ID

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

G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,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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