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G = C14xC4oD4order 224 = 25·7

Direct product of C14 and C4oD4

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

Aliases: C14xC4oD4, C28.55C23, C14.18C24, D4:3(C2xC14), (C2xD4):7C14, (C2xQ8):6C14, Q8:3(C2xC14), (D4xC14):16C2, (C22xC4):6C14, (Q8xC14):13C2, (C22xC28):13C2, (C2xC28):16C22, (C7xD4):12C22, (C2xC14).6C23, C2.3(C23xC14), C4.8(C22xC14), (C7xQ8):11C22, C23.11(C2xC14), C22.1(C22xC14), (C22xC14).30C22, (C2xC28)o(C7xD4), (C2xC28)o(C7xQ8), (C2xC4):5(C2xC14), (C2xC28)o(Q8xC14), SmallGroup(224,192)

Series: Derived Chief Lower central Upper central

C1C2 — C14xC4oD4
C1C2C14C2xC14C7xD4C7xC4oD4 — C14xC4oD4
C1C2 — C14xC4oD4
C1C2xC28 — C14xC4oD4

Generators and relations for C14xC4oD4
 G = < a,b,c,d | a14=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, Q8, C23, C14, C14, C14, C22xC4, C2xD4, C2xQ8, C4oD4, C28, C2xC14, C2xC14, C2xC14, C2xC4oD4, C2xC28, C2xC28, C7xD4, C7xQ8, C22xC14, C22xC28, D4xC14, Q8xC14, C7xC4oD4, C14xC4oD4
Quotients: C1, C2, C22, C7, C23, C14, C4oD4, C24, C2xC14, C2xC4oD4, C22xC14, C7xC4oD4, C23xC14, C14xC4oD4

Smallest permutation representation of C14xC4oD4
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 59 109 45)(2 60 110 46)(3 61 111 47)(4 62 112 48)(5 63 99 49)(6 64 100 50)(7 65 101 51)(8 66 102 52)(9 67 103 53)(10 68 104 54)(11 69 105 55)(12 70 106 56)(13 57 107 43)(14 58 108 44)(15 72 42 97)(16 73 29 98)(17 74 30 85)(18 75 31 86)(19 76 32 87)(20 77 33 88)(21 78 34 89)(22 79 35 90)(23 80 36 91)(24 81 37 92)(25 82 38 93)(26 83 39 94)(27 84 40 95)(28 71 41 96)
(1 32 109 19)(2 33 110 20)(3 34 111 21)(4 35 112 22)(5 36 99 23)(6 37 100 24)(7 38 101 25)(8 39 102 26)(9 40 103 27)(10 41 104 28)(11 42 105 15)(12 29 106 16)(13 30 107 17)(14 31 108 18)(43 74 57 85)(44 75 58 86)(45 76 59 87)(46 77 60 88)(47 78 61 89)(48 79 62 90)(49 80 63 91)(50 81 64 92)(51 82 65 93)(52 83 66 94)(53 84 67 95)(54 71 68 96)(55 72 69 97)(56 73 70 98)
(1 39)(2 40)(3 41)(4 42)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(43 81)(44 82)(45 83)(46 84)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 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), (1,59,109,45)(2,60,110,46)(3,61,111,47)(4,62,112,48)(5,63,99,49)(6,64,100,50)(7,65,101,51)(8,66,102,52)(9,67,103,53)(10,68,104,54)(11,69,105,55)(12,70,106,56)(13,57,107,43)(14,58,108,44)(15,72,42,97)(16,73,29,98)(17,74,30,85)(18,75,31,86)(19,76,32,87)(20,77,33,88)(21,78,34,89)(22,79,35,90)(23,80,36,91)(24,81,37,92)(25,82,38,93)(26,83,39,94)(27,84,40,95)(28,71,41,96), (1,32,109,19)(2,33,110,20)(3,34,111,21)(4,35,112,22)(5,36,99,23)(6,37,100,24)(7,38,101,25)(8,39,102,26)(9,40,103,27)(10,41,104,28)(11,42,105,15)(12,29,106,16)(13,30,107,17)(14,31,108,18)(43,74,57,85)(44,75,58,86)(45,76,59,87)(46,77,60,88)(47,78,61,89)(48,79,62,90)(49,80,63,91)(50,81,64,92)(51,82,65,93)(52,83,66,94)(53,84,67,95)(54,71,68,96)(55,72,69,97)(56,73,70,98), (1,39)(2,40)(3,41)(4,42)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(43,81)(44,82)(45,83)(46,84)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)>;

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

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

C14xC4oD4 is a maximal subgroup of
C4oD4:Dic7  C28.(C2xD4)  (D4xC14).11C4  (D4xC14):9C4  (D4xC14).16C4  (C7xD4):14D4  (C7xD4).32D4  (D4xC14):10C4  C28.76C24  C28.C24  C14.1042- 1+4  C14.1052- 1+4  C14.1062- 1+4  (C2xC28):15D4  C14.1452+ 1+4  C14.1462+ 1+4  C14.1072- 1+4  (C2xC28):17D4  C14.1082- 1+4  C14.1482+ 1+4  C14.C25
C14xC4oD4 is a maximal quotient of
D4xC2xC28  Q8xC2xC28

140 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J7A···7F14A···14R14S···14BB28A···28X28Y···28BH
order12222···244444···47···714···1414···1428···2828···28
size11112···211112···21···11···12···21···12···2

140 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C2C7C14C14C14C14C4oD4C7xC4oD4
kernelC14xC4oD4C22xC28D4xC14Q8xC14C7xC4oD4C2xC4oD4C22xC4C2xD4C2xQ8C4oD4C14C2
# reps1331861818648424

Matrix representation of C14xC4oD4 in GL3(F29) generated by

2800
0200
0020
,
100
0170
0017
,
2800
0271
0242
,
2800
0228
0327
G:=sub<GL(3,GF(29))| [28,0,0,0,20,0,0,0,20],[1,0,0,0,17,0,0,0,17],[28,0,0,0,27,24,0,1,2],[28,0,0,0,2,3,0,28,27] >;

C14xC4oD4 in GAP, Magma, Sage, TeX

C_{14}\times C_4\circ D_4
% in TeX

G:=Group("C14xC4oD4");
// GroupNames label

G:=SmallGroup(224,192);
// by ID

G=gap.SmallGroup(224,192);
# by ID

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

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

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