Copied to
clipboard

G = (D4xC14).16C4order 448 = 26·7

10th non-split extension by D4xC14 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D4xC14).16C4, (C2xC28).199D4, C28.214(C2xD4), (C22xC28).8C4, (C2xD4).9Dic7, (C2xD4).203D14, C28.D4:13C2, (C2xQ8).172D14, C28.10D4:13C2, C23.3(C2xDic7), (C22xC4).6Dic7, C28.91(C22:C4), (C2xC28).483C23, (C22xC4).163D14, C4.34(C23.D7), (D4xC14).244C22, (Q8xC14).207C22, C4.Dic7.47C22, C22.7(C23.D7), C22.8(C22xDic7), (C22xC28).209C22, C7:4(M4(2).8C22), (C2xC4oD4).6D7, C4.96(C2xC7:D4), (C2xC28).16(C2xC4), (C14xC4oD4).6C2, (C2xC4).5(C2xDic7), (C2xC4).91(C7:D4), C14.86(C2xC22:C4), (C2xC4.Dic7):23C2, C2.22(C2xC23.D7), (C22xC14).75(C2xC4), (C2xC4).131(C22xD7), (C2xC14).28(C22:C4), (C2xC14).201(C22xC4), SmallGroup(448,771)

Series: Derived Chief Lower central Upper central

C1C2xC14 — (D4xC14).16C4
C1C7C14C28C2xC28C4.Dic7C2xC4.Dic7 — (D4xC14).16C4
C7C14C2xC14 — (D4xC14).16C4
C1C4C22xC4C2xC4oD4

Generators and relations for (D4xC14).16C4
 G = < a,b,c,d | a14=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a7b, dcd-1=b2c >

Subgroups: 404 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C14, C14, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C28, C28, C28, C2xC14, C2xC14, C2xC14, C4.D4, C4.10D4, C2xM4(2), C2xC4oD4, C7:C8, C2xC28, C2xC28, C2xC28, C7xD4, C7xQ8, C22xC14, C22xC14, M4(2).8C22, C2xC7:C8, C4.Dic7, C4.Dic7, C22xC28, C22xC28, D4xC14, D4xC14, Q8xC14, C7xC4oD4, C28.D4, C28.10D4, C2xC4.Dic7, C14xC4oD4, (D4xC14).16C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22:C4, C22xC4, C2xD4, Dic7, D14, C2xC22:C4, C2xDic7, C7:D4, C22xD7, M4(2).8C22, C23.D7, C22xDic7, C2xC7:D4, C2xC23.D7, (D4xC14).16C4

Smallest permutation representation of (D4xC14).16C4
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 44 12 19)(2 45 13 20)(3 46 14 21)(4 47 8 15)(5 48 9 16)(6 49 10 17)(7 43 11 18)(22 55 35 41)(23 56 29 42)(24 50 30 36)(25 51 31 37)(26 52 32 38)(27 53 33 39)(28 54 34 40)(57 81 64 74)(58 82 65 75)(59 83 66 76)(60 84 67 77)(61 71 68 78)(62 72 69 79)(63 73 70 80)(85 102 92 109)(86 103 93 110)(87 104 94 111)(88 105 95 112)(89 106 96 99)(90 107 97 100)(91 108 98 101)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 43)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 42)(30 36)(31 37)(32 38)(33 39)(34 40)(35 41)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 71)(69 72)(70 73)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)
(1 63 41 110 12 70 55 103)(2 69 42 102 13 62 56 109)(3 61 36 108 14 68 50 101)(4 67 37 100 8 60 51 107)(5 59 38 106 9 66 52 99)(6 65 39 112 10 58 53 105)(7 57 40 104 11 64 54 111)(15 84 25 90 47 77 31 97)(16 76 26 96 48 83 32 89)(17 82 27 88 49 75 33 95)(18 74 28 94 43 81 34 87)(19 80 22 86 44 73 35 93)(20 72 23 92 45 79 29 85)(21 78 24 98 46 71 30 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), (1,44,12,19)(2,45,13,20)(3,46,14,21)(4,47,8,15)(5,48,9,16)(6,49,10,17)(7,43,11,18)(22,55,35,41)(23,56,29,42)(24,50,30,36)(25,51,31,37)(26,52,32,38)(27,53,33,39)(28,54,34,40)(57,81,64,74)(58,82,65,75)(59,83,66,76)(60,84,67,77)(61,71,68,78)(62,72,69,79)(63,73,70,80)(85,102,92,109)(86,103,93,110)(87,104,94,111)(88,105,95,112)(89,106,96,99)(90,107,97,100)(91,108,98,101), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,43)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,42)(30,36)(31,37)(32,38)(33,39)(34,40)(35,41)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108), (1,63,41,110,12,70,55,103)(2,69,42,102,13,62,56,109)(3,61,36,108,14,68,50,101)(4,67,37,100,8,60,51,107)(5,59,38,106,9,66,52,99)(6,65,39,112,10,58,53,105)(7,57,40,104,11,64,54,111)(15,84,25,90,47,77,31,97)(16,76,26,96,48,83,32,89)(17,82,27,88,49,75,33,95)(18,74,28,94,43,81,34,87)(19,80,22,86,44,73,35,93)(20,72,23,92,45,79,29,85)(21,78,24,98,46,71,30,91)>;

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

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

82 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G7A7B7C8A···8H14A···14I14J···14AA28A···28L28M···28AD
order122222244444447778···814···1414···1428···2828···28
size1122244112224422228···282···24···42···24···4

82 irreducible representations

dim11111112222222244
type+++++++-+-++
imageC1C2C2C2C2C4C4D4D7Dic7D14Dic7D14D14C7:D4M4(2).8C22(D4xC14).16C4
kernel(D4xC14).16C4C28.D4C28.10D4C2xC4.Dic7C14xC4oD4C22xC28D4xC14C2xC28C2xC4oD4C22xC4C22xC4C2xD4C2xD4C2xQ8C2xC4C7C1
# reps1222144436363324212

Matrix representation of (D4xC14).16C4 in GL4(F113) generated by

85000
08500
001090
24890109
,
09800
98000
1112981
00015
,
09800
15000
112115112
03011198
,
0010
1598115
0100
300015
G:=sub<GL(4,GF(113))| [85,0,0,24,0,85,0,89,0,0,109,0,0,0,0,109],[0,98,1,0,98,0,112,0,0,0,98,0,0,0,1,15],[0,15,112,0,98,0,1,30,0,0,15,111,0,0,112,98],[0,15,0,30,0,98,1,0,1,1,0,0,0,15,0,15] >;

(D4xC14).16C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})._{16}C_4
% in TeX

G:=Group("(D4xC14).16C4");
// GroupNames label

G:=SmallGroup(448,771);
// by ID

G=gap.SmallGroup(448,771);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,d*b*d^-1=a^7*b,d*c*d^-1=b^2*c>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<