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G = C4xC8oD4order 128 = 27

Direct product of C4 and C8oD4

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

Aliases: C4xC8oD4, D4.4C42, Q8.4C42, C42.589C23, D4o(C4xC8), C8o(C4xD4), C8o(C4xQ8), Q8o(C4xC8), C8o2(C8oD4), C8o(C4xM4(2)), M4(2)o(C4xC8), (C4xD4).41C4, (C4xQ8).38C4, C4.12(C2xC42), C8.48(C22xC4), C4.60(C23xC4), M4(2):24(C2xC4), (C4xM4(2)):44C2, (C2xC8).638C23, (C2xC4).624C24, C42.280(C2xC4), (C4xC8).435C22, C22.2(C2xC42), C4o2(C8o2M4(2)), C8o2(C8o2M4(2)), C8o2M4(2):39C2, C22.35(C23xC4), C2.16(C22xC42), C8:C4.172C22, C42o(C8o2M4(2)), C23.136(C22xC4), (C22xC8).584C22, (C2xC42).1101C22, (C22xC4).1489C23, C42:C2.347C22, (C2xM4(2)).381C22, (C2xC4xC8):41C2, C8o(C4xC4oD4), C4oD4o(C4xC8), C8o(C2xC8oD4), (C4xC8)o(C4xD4), (C4xC8)o(C4xQ8), (C2xQ8)o(C4xC8), (C2xC8):33(C2xC4), (C2xC8)o(C8oD4), C2.2(C2xC8oD4), C42o(C2xC8oD4), (C4xC8)o(C4xM4(2)), (C4xC8)o2(C8:C4), C4:C4.244(C2xC4), (C4xC4oD4).32C2, C4oD4.36(C2xC4), (C2xC8oD4).23C2, (C2xC8)o2(C4xM4(2)), (C4xC8)o2(C2xM4(2)), (C4xC8)o(C42:C2), (C2xD4).245(C2xC4), C22:C4.87(C2xC4), (C2xQ8).221(C2xC4), (C4xC8)o(C8o2M4(2)), (C22xC4).382(C2xC4), (C2xC4).454(C22xC4), (C2xC4oD4).337C22, (C2xC8)o(C4xC4oD4), (C4xC8)o(C2xC4oD4), (C4xC8)o(C2xC8oD4), SmallGroup(128,1606)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4xC8oD4
C1C2C22C2xC4C22xC4C2xC4oD4C4xC4oD4 — C4xC8oD4
C1C2 — C4xC8oD4
C1C4xC8 — C4xC8oD4
C1C2C2C2xC4 — C4xC8oD4

Generators and relations for C4xC8oD4
 G = < a,b,c,d | a4=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 316 in 286 conjugacy classes, 256 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, C4xC8, C4xC8, C8:C4, C2xC42, C42:C2, C4xD4, C4xQ8, C22xC8, C2xM4(2), C8oD4, C2xC4oD4, C2xC4xC8, C4xM4(2), C8o2M4(2), C4xC4oD4, C2xC8oD4, C4xC8oD4
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, C24, C2xC42, C8oD4, C23xC4, C22xC42, C2xC8oD4, C4xC8oD4

Smallest permutation representation of C4xC8oD4
On 64 points
Generators in S64
(1 33 63 47)(2 34 64 48)(3 35 57 41)(4 36 58 42)(5 37 59 43)(6 38 60 44)(7 39 61 45)(8 40 62 46)(9 18 32 49)(10 19 25 50)(11 20 26 51)(12 21 27 52)(13 22 28 53)(14 23 29 54)(15 24 30 55)(16 17 31 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)(57 58 59 60 61 62 63 64)
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(17 50 21 54)(18 51 22 55)(19 52 23 56)(20 53 24 49)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D···2I4A···4L4M···4AD8A···8P8Q···8AN
order12222···24···44···48···88···8
size11112···21···12···21···12···2

80 irreducible representations

dim1111111112
type++++++
imageC1C2C2C2C2C2C4C4C4C8oD4
kernelC4xC8oD4C2xC4xC8C4xM4(2)C8o2M4(2)C4xC4oD4C2xC8oD4C4xD4C4xQ8C8oD4C4
# reps1336121243216

Matrix representation of C4xC8oD4 in GL3(F17) generated by

1300
010
001
,
100
020
002
,
100
0130
004
,
100
004
0130
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,2],[1,0,0,0,13,0,0,0,4],[1,0,0,0,0,13,0,4,0] >;

C4xC8oD4 in GAP, Magma, Sage, TeX

C_4\times C_8\circ D_4
% in TeX

G:=Group("C4xC8oD4");
// GroupNames label

G:=SmallGroup(128,1606);
// by ID

G=gap.SmallGroup(128,1606);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,521,172]);
// Polycyclic

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

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