Copied to
clipboard

## G = C7×2- 1+4order 224 = 25·7

### Direct product of C7 and 2- 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×2- 1+4
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — C7×C4○D4 — C7×2- 1+4
 Lower central C1 — C2 — C7×2- 1+4
 Upper central C1 — C14 — C7×2- 1+4

Generators and relations for C7×2- 1+4
G = < a,b,c,d,e | a7=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C14, C2×Q8, C4○D4, C28, C2×C14, 2- 1+4, C2×C28, C7×D4, C7×Q8, Q8×C14, C7×C4○D4, C7×2- 1+4
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2- 1+4, C22×C14, C23×C14, C7×2- 1+4

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

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

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

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

C7×2- 1+4 is a maximal subgroup of   2- 1+4⋊D7  2- 1+4.D7  D28.34C23  D28.35C23  D28.39C23
C7×2- 1+4 is a maximal quotient of   C7×D4×Q8

119 conjugacy classes

 class 1 2A 2B ··· 2F 4A ··· 4J 7A ··· 7F 14A ··· 14F 14G ··· 14AJ 28A ··· 28BH order 1 2 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

119 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + - image C1 C2 C2 C7 C14 C14 2- 1+4 C7×2- 1+4 kernel C7×2- 1+4 Q8×C14 C7×C4○D4 2- 1+4 C2×Q8 C4○D4 C7 C1 # reps 1 5 10 6 30 60 1 6

Matrix representation of C7×2- 1+4 in GL4(𝔽29) generated by

 25 0 0 0 0 25 0 0 0 0 25 0 0 0 0 25
,
 0 1 0 0 28 0 0 0 0 0 0 28 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 28 0 0 27 0 28 27 0 0 1 1 0 1 0 0 1
,
 20 0 0 22 0 20 22 0 0 20 9 0 20 0 0 9
G:=sub<GL(4,GF(29))| [25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[0,28,0,0,1,0,0,0,0,0,0,1,0,0,28,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[28,0,0,1,0,28,1,0,0,27,1,0,27,0,0,1],[20,0,0,20,0,20,20,0,0,22,9,0,22,0,0,9] >;

C7×2- 1+4 in GAP, Magma, Sage, TeX

C_7\times 2_-^{1+4}
% in TeX

G:=Group("C7xES-(2,2)");
// GroupNames label

G:=SmallGroup(224,194);
// by ID

G=gap.SmallGroup(224,194);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-2,1369,679,1052,518,2883]);
// Polycyclic

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

׿
×
𝔽