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## G = C7×D42order 448 = 26·7

### Direct product of C7, D4 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×D42
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C22≀C2 — C7×D42
 Lower central C1 — C22 — C7×D42
 Upper central C1 — C2×C14 — C7×D42

Generators and relations for C7×D42
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C4×D4, C22≀C2, C4⋊D4, C41D4, C22×D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, D42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C14, C23×C14, D4×C28, C7×C22≀C2, C7×C4⋊D4, C7×C41D4, D4×C2×C14, C7×D42
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, 2+ 1+4, C7×D4, C22×C14, D42, D4×C14, C23×C14, D4×C2×C14, C7×2+ 1+4, C7×D42

Smallest permutation representation of C7×D42
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 35 67)(2 51 29 68)(3 52 30 69)(4 53 31 70)(5 54 32 64)(6 55 33 65)(7 56 34 66)(8 92 21 82)(9 93 15 83)(10 94 16 84)(11 95 17 78)(12 96 18 79)(13 97 19 80)(14 98 20 81)(22 100 111 89)(23 101 112 90)(24 102 106 91)(25 103 107 85)(26 104 108 86)(27 105 109 87)(28 99 110 88)(36 60 44 71)(37 61 45 72)(38 62 46 73)(39 63 47 74)(40 57 48 75)(41 58 49 76)(42 59 43 77)
(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)
(1 102 47 78)(2 103 48 79)(3 104 49 80)(4 105 43 81)(5 99 44 82)(6 100 45 83)(7 101 46 84)(8 54 110 71)(9 55 111 72)(10 56 112 73)(11 50 106 74)(12 51 107 75)(13 52 108 76)(14 53 109 77)(15 65 22 61)(16 66 23 62)(17 67 24 63)(18 68 25 57)(19 69 26 58)(20 70 27 59)(21 64 28 60)(29 85 40 96)(30 86 41 97)(31 87 42 98)(32 88 36 92)(33 89 37 93)(34 90 38 94)(35 91 39 95)
(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 61)(16 62)(17 63)(18 57)(19 58)(20 59)(21 60)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 64)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 81)(44 82)(45 83)(46 84)(47 78)(48 79)(49 80)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)

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

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

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

175 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E ··· 4I 7A ··· 7F 14A ··· 14R 14S ··· 14BN 14BO ··· 14CL 28A ··· 28X 28Y ··· 28BB order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

175 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 C7×D4 2+ 1+4 C7×2+ 1+4 kernel C7×D42 D4×C28 C7×C22≀C2 C7×C4⋊D4 C7×C4⋊1D4 D4×C2×C14 D42 C4×D4 C22≀C2 C4⋊D4 C4⋊1D4 C22×D4 C7×D4 D4 C14 C2 # reps 1 2 4 4 1 4 6 12 24 24 6 24 8 48 1 6

Matrix representation of C7×D42 in GL4(𝔽29) generated by

 20 0 0 0 0 20 0 0 0 0 16 0 0 0 0 16
,
 28 0 0 0 0 28 0 0 0 0 0 28 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 28
,
 28 2 0 0 28 1 0 0 0 0 28 0 0 0 0 28
,
 1 27 0 0 0 28 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,16,0,0,0,0,16],[28,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[28,28,0,0,2,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,27,28,0,0,0,0,28,0,0,0,0,28] >;

C7×D42 in GAP, Magma, Sage, TeX

C_7\times D_4^2
% in TeX

G:=Group("C7xD4^2");
// GroupNames label

G:=SmallGroup(448,1328);
// by ID

G=gap.SmallGroup(448,1328);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,1690]);
// Polycyclic

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

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