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G = D7xC8:C22  order 448 = 26·7

Direct product of D7 and C8:C22

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

Aliases: D7xC8:C22, C56:C23, D8:3D14, SD16:1D14, D56:1C22, D28:3C23, M4(2):7D14, C28.19C24, Dic14:3C23, (D7xD8):1C2, C7:C8:3C23, C4oD4:9D14, C8:1(C22xD7), (C2xD4):29D14, D8:D7:1C2, C8:D14:1C2, D56:C2:1C2, D4:D7:5C22, (D7xSD16):1C2, (C4xD7).42D4, C4.189(D4xD7), D4:3(C22xD7), (D4xD7):8C22, (C7xD8):1C22, (C7xD4):3C23, (C8xD7):1C22, Q8:D7:4C22, D4:D14:9C2, (Q8xD7):9C22, Q8:3(C22xD7), (C7xQ8):3C23, D14.67(C2xD4), C28.240(C2xD4), C4oD28:7C22, C56:C2:1C22, C8:D7:1C22, (D7xM4(2)):1C2, D4.D7:4C22, C4.19(C23xD7), C22.46(D4xD7), D4.D14:9C2, (D4xC14):21C22, D4:2D7:9C22, (C2xD28):35C22, Dic7.59(C2xD4), (C2xDic7).81D4, Q8:2D7:9C22, (C7xSD16):1C22, (C4xD7).12C23, (C2xC28).110C23, (C22xD7).101D4, C14.120(C22xD4), (C7xM4(2)):1C22, C4.Dic7:12C22, (C2xD4xD7):24C2, C7:4(C2xC8:C22), C2.93(C2xD4xD7), (D7xC4oD4):3C2, (C7xC8:C22):1C2, (C2xC14).65(C2xD4), (C7xC4oD4):5C22, (C2xC4xD7).160C22, (C2xC4).94(C22xD7), SmallGroup(448,1225)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — D7xC8:C22
Chief series
C1 — C7 — C14 — C28 — C4xD7 — C2xC4xD7 — C2xD4xD7 — D7xC8:C22
Lower central
C7 — C14 — C28 — D7xC8:C22
Upper central
C1 — C2 — C2xC4 — C8:C22

Generators and relations for D7xC8:C22
 G = < a,b,c,d,e | a7=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 1772 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2xC8, M4(2), M4(2), D8, D8, SD16, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, Dic7, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xM4(2), C2xD8, C2xSD16, C8:C22, C8:C22, C22xD4, C2xC4oD4, C7:C8, C56, Dic14, Dic14, C4xD7, C4xD7, D28, D28, D28, C2xDic7, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C7xD4, C7xD4, C7xQ8, C22xD7, C22xD7, C22xC14, C2xC8:C22, C8xD7, C8:D7, C56:C2, D56, C4.Dic7, D4:D7, D4.D7, Q8:D7, C7xM4(2), C7xD8, C7xSD16, C2xC4xD7, C2xC4xD7, C2xD28, C4oD28, C4oD28, D4xD7, D4xD7, D4xD7, D4:2D7, D4:2D7, Q8xD7, Q8:2D7, C2xC7:D4, D4xC14, C7xC4oD4, C23xD7, D7xM4(2), C8:D14, D7xD8, D8:D7, D7xSD16, D56:C2, D4.D14, D4:D14, C7xC8:C22, C2xD4xD7, D7xC4oD4, D7xC8:C22
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C24, D14, C8:C22, C22xD4, C22xD7, C2xC8:C22, D4xD7, C23xD7, C2xD4xD7, D7xC8:C22

Smallest permutation representation of D7xC8:C22
►On 56 points
Generators in S56
(1 37 55 14 17 45 27)(2 38 56 15 18 46 28)(3 39 49 16 19 47 29)(4 40 50 9 20 48 30)(5 33 51 10 21 41 31)(6 34 52 11 22 42 32)(7 35 53 12 23 43 25)(8 36 54 13 24 44 26)

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

(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 7)(3 5)(4 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)

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

55 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F7A7B7C8A8B8C8D14A14B14C14D14E14F14G···14O28A···28F28G28H28I56A···56F
order122222222222444444777888814141414141414···1428···2828282856···56
size11244477142828282241414282224428282224448···84···48888···8

55 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D7D14D14D14D14D14C8:C22D4xD7D4xD7D7xC8:C22
kernelD7xC8:C22D7xM4(2)C8:D14D7xD8D8:D7D7xSD16D56:C2D4.D14D4:D14C7xC8:C22C2xD4xD7D7xC4oD4C4xD7C2xDic7C22xD7C8:C22M4(2)D8SD16C2xD4C4oD4D7C4C22C1
# reps1112222111112113366332333

Matrix representation of D7xC8:C22 ►in GL8(F113)

791000000
1120000000
007910000
0011200000
00001000
00000100
00000010
00000001
,
8825000000
7925000000
0088250000
0079250000
0000112000
0000011200
0000001120
0000000112
,
00100000
00010000
1120000000
0112000000
000011222912
00000111172
00000111272
00001122201
,
1120000000
0112000000
00100000
00010000
00001122200
00000100
000041110
0000112022112
,
1120000000
0112000000
0011200000
0001120000
00001000
00000100
00007201120
00002910112

G:=sub<GL(8,GF(113))| [79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,112,0,0,0,0,22,1,1,22,0,0,0,0,91,111,112,0,0,0,0,0,2,72,72,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,41,112,0,0,0,0,22,1,1,0,0,0,0,0,0,0,1,22,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,72,2,0,0,0,0,0,1,0,91,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;
 

D7xC8:C22 in GAP, Magma, Sage, TeX 

D_7\times C_8\rtimes C_2^2
 
% in TeX
 
G:=Group("D7xC8:C2^2");
 
// GroupNames label
 
G:=SmallGroup(448,1225);
 
// by ID
 
G=gap.SmallGroup(448,1225);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,570,185,438,235,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^2=c^8=d^2=e^2=1,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,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
 
// generators/relations
 

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