Copied to
clipboard

G = C14xC4wrC2order 448 = 26·7

Direct product of C14 and C4wrC2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C14xC4wrC2, C4oD4:3C28, D4:5(C2xC28), (C2xD4):9C28, (C2xQ8):7C28, Q8:5(C2xC28), (D4xC14):21C4, (C2xC42):6C14, (Q8xC14):17C4, C4.72(D4xC14), (C4xC28):56C22, C42:15(C2xC14), (C2xC28).519D4, C28.477(C2xD4), C4.7(C22xC28), C23.39(C7xD4), M4(2):9(C2xC14), C22.12(D4xC14), (C14xM4(2)):30C2, (C2xM4(2)):12C14, C28.152(C22xC4), (C2xC28).896C23, (C22xC14).161D4, C28.117(C22:C4), (C7xM4(2)):38C22, (C22xC28).584C22, (C2xC4xC28):19C2, (C7xC4oD4):9C4, (C7xD4):25(C2xC4), (C7xQ8):23(C2xC4), (C2xC4).70(C7xD4), (C2xC4).50(C2xC28), C4oD4.6(C2xC14), (C2xC4oD4).6C14, C4.33(C7xC22:C4), (C2xC28).271(C2xC4), (C14xC4oD4).20C2, (C2xC14).407(C2xD4), C2.23(C14xC22:C4), C22.6(C7xC22:C4), C14.111(C2xC22:C4), (C2xC4).71(C22xC14), (C7xC4oD4).51C22, (C22xC4).113(C2xC14), (C2xC14).140(C22:C4), SmallGroup(448,828)

Series: Derived Chief Lower central Upper central

C1C4 — C14xC4wrC2
C1C2C4C2xC4C2xC28C7xM4(2)C7xC4wrC2 — C14xC4wrC2
C1C2C4 — C14xC4wrC2
C1C2xC28C22xC28 — C14xC4wrC2

Generators and relations for C14xC4wrC2
 G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C42, C42, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C28, C28, C2xC14, C2xC14, C4wrC2, C2xC42, C2xM4(2), C2xC4oD4, C56, C2xC28, C2xC28, C7xD4, C7xD4, C7xQ8, C7xQ8, C22xC14, C22xC14, C2xC4wrC2, C4xC28, C4xC28, C2xC56, C7xM4(2), C7xM4(2), C22xC28, C22xC28, D4xC14, D4xC14, Q8xC14, C7xC4oD4, C7xC4oD4, C7xC4wrC2, C2xC4xC28, C14xM4(2), C14xC4oD4, C14xC4wrC2
Quotients: C1, C2, C4, C22, C7, C2xC4, D4, C23, C14, C22:C4, C22xC4, C2xD4, C28, C2xC14, C4wrC2, C2xC22:C4, C2xC28, C7xD4, C22xC14, C2xC4wrC2, C7xC22:C4, C22xC28, D4xC14, C7xC4wrC2, C14xC22:C4, C14xC4wrC2

Smallest permutation representation of C14xC4wrC2
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 75 16 86)(2 76 17 87)(3 77 18 88)(4 78 19 89)(5 79 20 90)(6 80 21 91)(7 81 22 92)(8 82 23 93)(9 83 24 94)(10 84 25 95)(11 71 26 96)(12 72 27 97)(13 73 28 98)(14 74 15 85)(29 111 59 44)(30 112 60 45)(31 99 61 46)(32 100 62 47)(33 101 63 48)(34 102 64 49)(35 103 65 50)(36 104 66 51)(37 105 67 52)(38 106 68 53)(39 107 69 54)(40 108 70 55)(41 109 57 56)(42 110 58 43)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 99)(13 100)(14 101)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 43)(25 44)(26 45)(27 46)(28 47)(29 84)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)
(1 82 16 93)(2 83 17 94)(3 84 18 95)(4 71 19 96)(5 72 20 97)(6 73 21 98)(7 74 22 85)(8 75 23 86)(9 76 24 87)(10 77 25 88)(11 78 26 89)(12 79 27 90)(13 80 28 91)(14 81 15 92)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 99)(54 100)(55 101)(56 102)

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

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

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

196 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O4P7A···7F8A8B8C8D14A···14R14S···14AD14AE···14AP28A···28X28Y···28CF28CG···28CR56A···56X
order1222222244444···4447···7888814···1414···1414···1428···2828···2828···2856···56
size1111224411112···2441···144441···12···24···41···12···24···44···4

196 irreducible representations

dim1111111111111111222222
type+++++++
imageC1C2C2C2C2C4C4C4C7C14C14C14C14C28C28C28D4D4C4wrC2C7xD4C7xD4C7xC4wrC2
kernelC14xC4wrC2C7xC4wrC2C2xC4xC28C14xM4(2)C14xC4oD4D4xC14Q8xC14C7xC4oD4C2xC4wrC2C4wrC2C2xC42C2xM4(2)C2xC4oD4C2xD4C2xQ8C4oD4C2xC28C22xC14C14C2xC4C23C2
# reps1411122462466612122431818648

Matrix representation of C14xC4wrC2 in GL3(F113) generated by

11200
0640
0064
,
100
0150
0098
,
11200
0098
0150
,
100
0980
001
G:=sub<GL(3,GF(113))| [112,0,0,0,64,0,0,0,64],[1,0,0,0,15,0,0,0,98],[112,0,0,0,0,15,0,98,0],[1,0,0,0,98,0,0,0,1] >;

C14xC4wrC2 in GAP, Magma, Sage, TeX

C_{14}\times C_4\wr C_2
% in TeX

G:=Group("C14xC4wrC2");
// GroupNames label

G:=SmallGroup(448,828);
// by ID

G=gap.SmallGroup(448,828);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,9804,4911,172,124]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<