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G = D4xC2xC8order 128 = 27

Direct product of C2xC8 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4xC2xC8, C42.679C23, C8o(C8xD4), C4o(C8xD4), C8o2(C4xD4), C23:5(C2xC8), (C23xC8):7C2, C4:1(C22xC8), C4:C8:94C22, (C4xC8):69C22, (C4xD4).42C4, C4.182(C4xD4), C2.4(C23xC8), C24.99(C2xC4), C22:1(C22xC8), C22:C8:82C22, C42.281(C2xC4), (C2xC4).645C24, (C2xC8).476C23, (C22xC8):64C22, (C22xD4).46C4, C4.191(C22xD4), C22.112(C4xD4), (C4xD4).359C22, C22.40(C8oD4), C22.40(C23xC4), (C23xC4).659C22, C23.140(C22xC4), (C22xC4).1274C23, (C2xC42).1108C22, C8o2(C2xC4:C8), C8o2(C2xC4:C4), C4:C4o3(C2xC8), (C2xC4xC8):18C2, (C2xC8)o(C8xD4), (C2xC4)o(C8xD4), C2.4(C2xC4xD4), (C2xC4):8(C2xC8), (C2xC8)o2(C4xD4), C4:C4o(C22xC8), (C2xC4:C8):55C2, (C2xC8)o3(C4:C8), C22:C4o3(C2xC8), C8o2(C2xC22:C8), C4:C8o2(C22xC8), C8o2(C2xC22:C4), (C2xC4xD4).92C2, C2.3(C2xC8oD4), (C2xC8)o(C22xD4), (C4xD4)o(C22xC8), (C2xD4)o(C22xC8), (C2xC4:C4).84C4, C22:C4o(C22xC8), C4:C4.246(C2xC4), (C2xC8)o3(C22:C8), (C2xC22:C8):49C2, C4.296(C2xC4oD4), C22:C8o2(C22xC8), (C22xC8)o(C22xD4), (C2xD4).248(C2xC4), (C2xC4).1570(C2xD4), (C2xC22:C4).55C4, C22:C4.90(C2xC4), (C2xC4).955(C4oD4), (C2xC4).292(C22xC4), (C22xC4).383(C2xC4), (C2xC8)o(C2xC4xD4), (C2xC8)o2(C2xC4:C8), (C2xC8)o2(C2xC4:C4), (C22xC8)o(C2xC4xD4), (C2xC4:C4)o(C22xC8), (C22xC8)o(C2xC4:C8), (C2xC8)o2(C2xC22:C4), (C2xC8)o2(C2xC22:C8), (C2xC22:C4)o(C22xC8), (C22xC8)o(C2xC22:C8), SmallGroup(128,1658)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — D4xC2xC8
C1C2C4C2xC4C22xC4C22xC8C23xC8 — D4xC2xC8
C1C2 — D4xC2xC8
C1C22xC8 — D4xC2xC8
C1C2C2C2xC4 — D4xC2xC8

Generators and relations for D4xC2xC8
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 420 in 308 conjugacy classes, 196 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C2xD4, C24, C4xC8, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C22xC8, C22xC8, C22xC8, C23xC4, C22xD4, C2xC4xC8, C2xC22:C8, C2xC4:C8, C8xD4, C2xC4xD4, C23xC8, D4xC2xC8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C2xC8, C22xC4, C2xD4, C4oD4, C24, C4xD4, C22xC8, C8oD4, C23xC4, C22xD4, C2xC4oD4, C8xD4, C2xC4xD4, C23xC8, C2xC8oD4, D4xC2xC8

Smallest permutation representation of D4xC2xC8
On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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)
(1 31 43 54)(2 32 44 55)(3 25 45 56)(4 26 46 49)(5 27 47 50)(6 28 48 51)(7 29 41 52)(8 30 42 53)(9 33 22 62)(10 34 23 63)(11 35 24 64)(12 36 17 57)(13 37 18 58)(14 38 19 59)(15 39 20 60)(16 40 21 61)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)

G:=sub<Sym(64)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,31,43,54)(2,32,44,55)(3,25,45,56)(4,26,46,49)(5,27,47,50)(6,28,48,51)(7,29,41,52)(8,30,42,53)(9,33,22,62)(10,34,23,63)(11,35,24,64)(12,36,17,57)(13,37,18,58)(14,38,19,59)(15,39,20,60)(16,40,21,61), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)>;

G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,31,43,54)(2,32,44,55)(3,25,45,56)(4,26,46,49)(5,27,47,50)(6,28,48,51)(7,29,41,52)(8,30,42,53)(9,33,22,62)(10,34,23,63)(11,35,24,64)(12,36,17,57)(13,37,18,58)(14,38,19,59)(15,39,20,60)(16,40,21,61), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61) );

G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(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)], [(1,31,43,54),(2,32,44,55),(3,25,45,56),(4,26,46,49),(5,27,47,50),(6,28,48,51),(7,29,41,52),(8,30,42,53),(9,33,22,62),(10,34,23,63),(11,35,24,64),(12,36,17,57),(13,37,18,58),(14,38,19,59),(15,39,20,60),(16,40,21,61)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)]])

80 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4X8A···8P8Q···8AN
order12···22···24···44···48···88···8
size11···12···21···12···21···12···2

80 irreducible representations

dim111111111111222
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4C8D4C4oD4C8oD4
kernelD4xC2xC8C2xC4xC8C2xC22:C8C2xC4:C8C8xD4C2xC4xD4C23xC8C2xC22:C4C2xC4:C4C4xD4C22xD4C2xD4C2xC8C2xC4C22
# reps1121812428232448

Matrix representation of D4xC2xC8 in GL4(F17) generated by

16000
01600
0010
0001
,
15000
01600
00150
00015
,
16000
01600
00161
00151
,
1000
0100
00161
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,16,0,0,0,0,15,0,0,0,0,15],[16,0,0,0,0,16,0,0,0,0,16,15,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,1,1] >;

D4xC2xC8 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_8
% in TeX

G:=Group("D4xC2xC8");
// GroupNames label

G:=SmallGroup(128,1658);
// by ID

G=gap.SmallGroup(128,1658);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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