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G = D7xC4oD4  order 224 = 25·7

Direct product of D7 and C4oD4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D7xC4oD4, D4:7D14, Q8:6D14, D28:10C22, C28.25C23, C14.11C24, D14.6C23, Dic14:10C22, Dic7.10C23, (D4xD7):6C2, (C2xC4):7D14, (Q8xD7):6C2, C4oD28:7C2, D4:2D7:6C2, (C2xC28):4C22, Q8:2D7:6C2, (C7xD4):8C22, (C4xD7):7C22, C7:D4:4C22, (C7xQ8):7C22, (C2xC14).3C23, C4.25(C22xD7), C2.12(C23xD7), C22.2(C22xD7), (C2xDic7):10C22, (C22xD7).31C22, (C2xC4xD7):6C2, C7:4(C2xC4oD4), (C7xC4oD4):3C2, SmallGroup(224,184)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D7xC4oD4
Chief series
C1 — C7 — C14 — D14 — C22xD7 — C2xC4xD7 — D7xC4oD4
Lower central
C7 — C14 — D7xC4oD4
Upper central
C1 — C4 — C4oD4

Generators and relations for D7xC4oD4
 G = < a,b,c,d,e | a7=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 614 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, Dic7, Dic7, C28, C28, D14, D14, D14, C2xC14, C2xC4oD4, Dic14, C4xD7, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C7xQ8, C22xD7, C2xC4xD7, C4oD28, D4xD7, D4:2D7, Q8xD7, Q8:2D7, C7xC4oD4, D7xC4oD4
Quotients: C1, C2, C22, C23, D7, C4oD4, C24, D14, C2xC4oD4, C22xD7, C23xD7, D7xC4oD4

Smallest permutation representation of D7xC4oD4
►On 56 points
Generators in S56
(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)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)

(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)

(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)

(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
 
G:=sub<Sym(56)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)>;
 
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49), (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56) );
 
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53)], [(1,41,13,34),(2,42,14,35),(3,36,8,29),(4,37,9,30),(5,38,10,31),(6,39,11,32),(7,40,12,33),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49)], [(1,27,13,20),(2,28,14,21),(3,22,8,15),(4,23,9,16),(5,24,10,17),(6,25,11,18),(7,26,12,19),(29,50,36,43),(30,51,37,44),(31,52,38,45),(32,53,39,46),(33,54,40,47),(34,55,41,48),(35,56,42,49)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)]])
 

D7xC4oD4 is a maximal subgroup of
 C42:D14  C56.49C23  D8:10D14  SD16:D14  D56:C22  C14.C25  D14.C24  D28.39C23
D7xC4oD4 is a maximal quotient of
 C42.88D14  C42.188D14  C42:8D14  C42:10D14  C42.93D14  C42.94D14  C42.95D14  C42.96D14  C42.97D14  C42.98D14  C4xD4:2D7  C42.102D14  D4:5Dic14  C42.104D14  C4xD4xD7  C42:12D14  C42.228D14  D4:5D28  C42:16D14  C42.229D14  C42.113D14  C42.114D14  C42.122D14  Q8:5Dic14  C4xQ8xD7  C4xQ8:2D7  Q8:5D28  C42.232D14  C42.131D14  C42.132D14  C28:(C4oD4)  Dic14:20D4  C4:C4.178D14  C14.342+ 1+4  C4:C4:21D14  C14.402+ 1+4  D28:20D4  C14.422+ 1+4  C14.432+ 1+4  C14.442+ 1+4  C14.452+ 1+4  (Q8xDic7):C2  C22:Q8:25D7  C4:C4:26D14  D28:22D4  Dic14:22D4  C14.522+ 1+4  C14.532+ 1+4  C14.202- 1+4  C14.212- 1+4  C14.222- 1+4  C14.232- 1+4  C4:C4.197D14  C14.802- 1+4  C14.1212+ 1+4  C14.822- 1+4  C4:C4:28D14  C14.612+ 1+4  C14.1222+ 1+4  C14.622+ 1+4  C14.832- 1+4  C14.642+ 1+4  C14.842- 1+4  C14.662+ 1+4  C14.672+ 1+4  C42.233D14  C42.137D14  C42.138D14  C42.139D14  D28:10D4  Dic14:10D4  C42:20D14  C42:21D14  C42.234D14  C42.143D14  Dic14:7Q8  C42.236D14  D28:7Q8  C42.237D14  C42.150D14  C42.151D14  C42.152D14  C42.153D14  C42.154D14  C42.159D14  C42.160D14  C42:23D14  C42:24D14  C42.189D14  C42.161D14  C42.162D14  C42.163D14  C42.164D14  C14.1042- 1+4  (C2xC28):15D4  C14.1452+ 1+4  C14.1072- 1+4  (C2xC28):17D4  C14.1482+ 1+4

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J7A7B7C14A14B14C14D···14L28A···28F28G···28O
order1222222222444444444477714141414···1428···2828···28
size112227714141411222771414142222224···42···24···4

50 irreducible representations

dim11111111222224
type++++++++++++
imageC1C2C2C2C2C2C2C2D7C4oD4D14D14D14D7xC4oD4
kernelD7xC4oD4C2xC4xD7C4oD28D4xD7D4:2D7Q8xD7Q8:2D7C7xC4oD4C4oD4D7C2xC4D4Q8C1
# reps13333111349936

Matrix representation of D7xC4oD4 ►in GL4(F29) generated by

22100
272500
0010
0001
,
252800
15400
00280
00028
,
1000
0100
00170
00017
,
28000
02800
002822
00211
,
28000
02800
002822
0001
G:=sub<GL(4,GF(29))| [22,27,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[25,15,0,0,28,4,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[28,0,0,0,0,28,0,0,0,0,28,21,0,0,22,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,22,1] >;
 

D7xC4oD4 in GAP, Magma, Sage, TeX 

D_7\times C_4\circ D_4
 
% in TeX
 
G:=Group("D7xC4oD4");
 
// GroupNames label
 
G:=SmallGroup(224,184);
 
// by ID
 
G=gap.SmallGroup(224,184);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,297,6917]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
 
// generators/relations
 

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