Copied to
clipboard

## G = D7×C42⋊2C2order 448 = 26·7

### Direct product of D7 and C42⋊2C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C42⋊2C2
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — D7×C42⋊2C2
 Lower central C7 — C2×C14 — D7×C42⋊2C2
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for D7×C422C2
G = < a,b,c,d,e | a7=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 1164 in 246 conjugacy classes, 101 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C422C2, C422C2, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×C14, C2×C422C2, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C23×D7, D7×C42, C422D7, C23.D14, D7×C22⋊C4, D7×C4⋊C4, C4⋊C4⋊D7, C7×C422C2, D7×C422C2
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C422C2, C2×C4○D4, C22×D7, C2×C422C2, C23×D7, D7×C4○D4, D7×C422C2

Smallest permutation representation of D7×C422C2
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 19)(2 18)(3 17)(4 16)(5 15)(6 21)(7 20)(8 24)(9 23)(10 22)(11 28)(12 27)(13 26)(14 25)(29 45)(30 44)(31 43)(32 49)(33 48)(34 47)(35 46)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 53)(57 73)(58 72)(59 71)(60 77)(61 76)(62 75)(63 74)(64 80)(65 79)(66 78)(67 84)(68 83)(69 82)(70 81)(85 101)(86 100)(87 99)(88 105)(89 104)(90 103)(91 102)(92 108)(93 107)(94 106)(95 112)(96 111)(97 110)(98 109)
(1 97 13 90)(2 98 14 91)(3 92 8 85)(4 93 9 86)(5 94 10 87)(6 95 11 88)(7 96 12 89)(15 106 22 99)(16 107 23 100)(17 108 24 101)(18 109 25 102)(19 110 26 103)(20 111 27 104)(21 112 28 105)(29 71 36 78)(30 72 37 79)(31 73 38 80)(32 74 39 81)(33 75 40 82)(34 76 41 83)(35 77 42 84)(43 57 50 64)(44 58 51 65)(45 59 52 66)(46 60 53 67)(47 61 54 68)(48 62 55 69)(49 63 56 70)
(1 55 27 34)(2 56 28 35)(3 50 22 29)(4 51 23 30)(5 52 24 31)(6 53 25 32)(7 54 26 33)(8 43 15 36)(9 44 16 37)(10 45 17 38)(11 46 18 39)(12 47 19 40)(13 48 20 41)(14 49 21 42)(57 106 78 85)(58 107 79 86)(59 108 80 87)(60 109 81 88)(61 110 82 89)(62 111 83 90)(63 112 84 91)(64 99 71 92)(65 100 72 93)(66 101 73 94)(67 102 74 95)(68 103 75 96)(69 104 76 97)(70 105 77 98)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4F 4G 4H 4I 4J ··· 4O 4P 4Q 4R 7A 7B 7C 14A ··· 14I 14J 14K 14L 28A ··· 28R 28S ··· 28AA order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 ··· 4 4 4 4 7 7 7 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 7 7 7 7 28 2 ··· 2 4 4 4 14 ··· 14 28 28 28 2 2 2 2 ··· 2 8 8 8 4 ··· 4 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D7×C4○D4 kernel D7×C42⋊2C2 D7×C42 C42⋊2D7 C23.D14 D7×C22⋊C4 D7×C4⋊C4 C4⋊C4⋊D7 C7×C42⋊2C2 C42⋊2C2 D14 C42 C22⋊C4 C4⋊C4 C2 # reps 1 1 1 3 3 3 3 1 3 12 3 9 9 18

Matrix representation of D7×C422C2 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 1 0 0 0 0 27 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 10 28 0 0 0 0 12 19 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 12 27 0 0 0 0 28 17
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 24 0 0 0 0 12 28
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 28

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,27,0,0,0,0,1,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,12,0,0,0,0,28,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,28,0,0,0,0,27,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,24,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,28] >;

D7×C422C2 in GAP, Magma, Sage, TeX

D_7\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("D7xC4^2:2C2");
// GroupNames label

G:=SmallGroup(448,1156);
// by ID

G=gap.SmallGroup(448,1156);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,346,297,136,18822]);
// Polycyclic

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

׿
×
𝔽