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## G = C14×2+ 1+4order 448 = 26·7

### Direct product of C14 and 2+ 1+4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C14×2+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, Q8, C23, C23, C14, C14, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C28, C2×C14, C2×C14, C2×C14, C22×D4, C2×C4○D4, 2+ 1+4, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C2×2+ 1+4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×2+ 1+4, C14×2+ 1+4
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2+ 1+4, C25, C22×C14, C2×2+ 1+4, C23×C14, C7×2+ 1+4, C24×C14, C14×2+ 1+4

Smallest permutation representation of C14×2+ 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 45 28 57)(2 46 15 58)(3 47 16 59)(4 48 17 60)(5 49 18 61)(6 50 19 62)(7 51 20 63)(8 52 21 64)(9 53 22 65)(10 54 23 66)(11 55 24 67)(12 56 25 68)(13 43 26 69)(14 44 27 70)(29 106 75 95)(30 107 76 96)(31 108 77 97)(32 109 78 98)(33 110 79 85)(34 111 80 86)(35 112 81 87)(36 99 82 88)(37 100 83 89)(38 101 84 90)(39 102 71 91)(40 103 72 92)(41 104 73 93)(42 105 74 94)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 82)(16 83)(17 84)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 112)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)
(1 52 28 64)(2 53 15 65)(3 54 16 66)(4 55 17 67)(5 56 18 68)(6 43 19 69)(7 44 20 70)(8 45 21 57)(9 46 22 58)(10 47 23 59)(11 48 24 60)(12 49 25 61)(13 50 26 62)(14 51 27 63)(29 88 75 99)(30 89 76 100)(31 90 77 101)(32 91 78 102)(33 92 79 103)(34 93 80 104)(35 94 81 105)(36 95 82 106)(37 96 83 107)(38 97 84 108)(39 98 71 109)(40 85 72 110)(41 86 73 111)(42 87 74 112)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 85)(14 86)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 53)(30 54)(31 55)(32 56)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(57 81)(58 82)(59 83)(60 84)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)

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

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

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

238 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2U 4A ··· 4L 7A ··· 7F 14A ··· 14R 14S ··· 14DV 28A ··· 28BT order 1 2 2 2 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 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 2+ 1+4 C7×2+ 1+4 kernel C14×2+ 1+4 D4×C2×C14 C14×C4○D4 C7×2+ 1+4 C2×2+ 1+4 C22×D4 C2×C4○D4 2+ 1+4 C14 C2 # reps 1 9 6 16 6 54 36 96 2 12

Matrix representation of C14×2+ 1+4 in GL5(𝔽29)

 28 0 0 0 0 0 23 0 0 0 0 0 23 0 0 0 0 0 23 0 0 0 0 0 23
,
 1 0 0 0 0 0 1 0 0 2 0 1 0 1 1 0 0 28 0 28 0 28 0 0 28
,
 1 0 0 0 0 0 28 2 0 0 0 0 1 0 0 0 0 28 0 28 0 0 28 28 0
,
 1 0 0 0 0 0 1 0 0 2 0 0 0 28 1 0 28 1 0 28 0 28 0 0 28
,
 28 0 0 0 0 0 1 0 2 0 0 0 0 1 28 0 0 0 28 0 0 0 28 28 0

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,1,1,0,28,0,0,0,28,0,0,0,1,0,0,0,2,1,28,28],[1,0,0,0,0,0,28,0,0,0,0,2,1,28,28,0,0,0,0,28,0,0,0,28,0],[1,0,0,0,0,0,1,0,28,28,0,0,0,1,0,0,0,28,0,0,0,2,1,28,28],[28,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,2,1,28,28,0,0,28,0,0] >;

C14×2+ 1+4 in GAP, Magma, Sage, TeX

C_{14}\times 2_+^{1+4}
% in TeX

G:=Group("C14xES+(2,2)");
// GroupNames label

G:=SmallGroup(448,1389);
// by ID

G=gap.SmallGroup(448,1389);
# by ID

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

G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^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=b^2*d>;
// generators/relations

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