Copied to
clipboard

G = C7×C2.C25order 448 = 26·7

Direct product of C7 and C2.C25

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C2.C25
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — D4×C14 — C7×2+ 1+4 — C7×C2.C25
 Lower central C1 — C2 — C7×C2.C25
 Upper central C1 — C28 — C7×C2.C25

Generators and relations for C7×C2.C25
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 930 in 810 conjugacy classes, 750 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, D4, Q8, C23, C14, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C28, C7×D4, C7×Q8, C22×C14, C2.C25, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4○D4, C7×2+ 1+4, C7×2- 1+4, C7×C2.C25
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, C25, C22×C14, C2.C25, C23×C14, C24×C14, C7×C2.C25

Smallest permutation representation of C7×C2.C25
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 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 111)(9 112)(10 106)(11 107)(12 108)(13 109)(14 110)(22 34)(23 35)(24 29)(25 30)(26 31)(27 32)(28 33)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)(50 62)(51 63)(52 57)(53 58)(54 59)(55 60)(56 61)(64 76)(65 77)(66 71)(67 72)(68 73)(69 74)(70 75)(78 90)(79 91)(80 85)(81 86)(82 87)(83 88)(84 89)(92 104)(93 105)(94 99)(95 100)(96 101)(97 102)(98 103)
(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)(50 62)(51 63)(52 57)(53 58)(54 59)(55 60)(56 61)(64 76)(65 77)(66 71)(67 72)(68 73)(69 74)(70 75)(78 90)(79 91)(80 85)(81 86)(82 87)(83 88)(84 89)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 64)(7 65)(8 60)(9 61)(10 62)(11 63)(12 57)(13 58)(14 59)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(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 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(8 111)(9 112)(10 106)(11 107)(12 108)(13 109)(14 110)(64 76)(65 77)(66 71)(67 72)(68 73)(69 74)(70 75)(78 90)(79 91)(80 85)(81 86)(82 87)(83 88)(84 89)(92 104)(93 105)(94 99)(95 100)(96 101)(97 102)(98 103)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 88)(9 89)(10 90)(11 91)(12 85)(13 86)(14 87)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 29 15 24)(2 30 16 25)(3 31 17 26)(4 32 18 27)(5 33 19 28)(6 34 20 22)(7 35 21 23)(8 102 111 97)(9 103 112 98)(10 104 106 92)(11 105 107 93)(12 99 108 94)(13 100 109 95)(14 101 110 96)(36 62 48 50)(37 63 49 51)(38 57 43 52)(39 58 44 53)(40 59 45 54)(41 60 46 55)(42 61 47 56)(64 90 76 78)(65 91 77 79)(66 85 71 80)(67 86 72 81)(68 87 73 82)(69 88 74 83)(70 89 75 84)

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

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

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

238 conjugacy classes

 class 1 2A 2B ··· 2P 4A 4B 4C ··· 4Q 7A ··· 7F 14A ··· 14F 14G ··· 14CR 28A ··· 28L 28M ··· 28CX order 1 2 2 ··· 2 4 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

238 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C7 C14 C14 C14 C2.C25 C7×C2.C25 kernel C7×C2.C25 C14×C4○D4 C7×2+ 1+4 C7×2- 1+4 C2.C25 C2×C4○D4 2+ 1+4 2- 1+4 C7 C1 # reps 1 15 10 6 6 90 60 36 2 12

Matrix representation of C7×C2.C25 in GL4(𝔽29) generated by

 20 0 0 0 0 20 0 0 0 0 20 0 0 0 0 20
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 28 0 0 0 0 28 0 0 0 0 1
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,28,0,0,0,0,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;

C7×C2.C25 in GAP, Magma, Sage, TeX

C_7\times C_2.C_2^5
% in TeX

G:=Group("C7xC2.C2^5");
// GroupNames label

G:=SmallGroup(448,1391);
// by ID

G=gap.SmallGroup(448,1391);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-7,-2,3165,2403,6499,522]);
// Polycyclic

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

׿
×
𝔽