Copied to
clipboard

## G = D7×C42order 224 = 25·7

### Direct product of C42 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D7×C42
 Lower central C7 — D7×C42
 Upper central C1 — C42

Generators and relations for D7×C42
G = < a,b,c,d | a4=b4=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 342 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, D7, C14, C42, C42, C22×C4, Dic7, C28, D14, C2×C14, C2×C42, C4×D7, C2×Dic7, C2×C28, C22×D7, C4×Dic7, C4×C28, C2×C4×D7, D7×C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, D14, C2×C42, C4×D7, C22×D7, C2×C4×D7, D7×C42

Smallest permutation representation of D7×C42
On 112 points
Generators in S112
(1 69 13 62)(2 70 14 63)(3 64 8 57)(4 65 9 58)(5 66 10 59)(6 67 11 60)(7 68 12 61)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 55 20 41)(2 56 21 42)(3 50 15 36)(4 51 16 37)(5 52 17 38)(6 53 18 39)(7 54 19 40)(8 43 22 29)(9 44 23 30)(10 45 24 31)(11 46 25 32)(12 47 26 33)(13 48 27 34)(14 49 28 35)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)(64 73)(65 72)(66 71)(67 77)(68 76)(69 75)(70 74)(85 108)(86 107)(87 106)(88 112)(89 111)(90 110)(91 109)(92 101)(93 100)(94 99)(95 105)(96 104)(97 103)(98 102)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 7A 7B 7C 14A ··· 14I 28A ··· 28AJ order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 7 7 7 7 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D7 D14 C4×D7 kernel D7×C42 C4×Dic7 C4×C28 C2×C4×D7 C4×D7 C42 C2×C4 C4 # reps 1 3 1 3 24 3 9 36

Matrix representation of D7×C42 in GL3(𝔽29) generated by

 1 0 0 0 17 0 0 0 17
,
 17 0 0 0 28 0 0 0 28
,
 1 0 0 0 0 1 0 28 18
,
 1 0 0 0 0 28 0 28 0
G:=sub<GL(3,GF(29))| [1,0,0,0,17,0,0,0,17],[17,0,0,0,28,0,0,0,28],[1,0,0,0,0,28,0,1,18],[1,0,0,0,0,28,0,28,0] >;

D7×C42 in GAP, Magma, Sage, TeX

D_7\times C_4^2
% in TeX

G:=Group("D7xC4^2");
// GroupNames label

G:=SmallGroup(224,66);
// by ID

G=gap.SmallGroup(224,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,50,6917]);
// Polycyclic

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

׿
×
𝔽