Copied to
clipboard

## G = C2×C4⋊D8order 128 = 27

### Direct product of C2 and C4⋊D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4⋊D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C2×C4⋊D8
 Lower central C1 — C2 — C2×C4 — C2×C4⋊D8
 Upper central C1 — C23 — C2×C42 — C2×C4⋊D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4⋊D8

Generators and relations for C2×C4⋊D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 780 in 324 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C41D4, C41D4, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C22×D4, C2×D4⋊C4, C2×C4⋊C8, C4⋊D8, C2×C4×D4, C2×C41D4, C22×D8, C2×C4⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C4⋊D4, C2×D8, C8⋊C22, C22×D4, C2×C4○D4, C4⋊D8, C2×C4⋊D4, C22×D8, C2×C8⋊C22, C2×C4⋊D8

Smallest permutation representation of C2×C4⋊D8
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 36 47 61)(2 62 48 37)(3 38 41 63)(4 64 42 39)(5 40 43 57)(6 58 44 33)(7 34 45 59)(8 60 46 35)(9 27 18 55)(10 56 19 28)(11 29 20 49)(12 50 21 30)(13 31 22 51)(14 52 23 32)(15 25 24 53)(16 54 17 26)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 56)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(41 45)(42 44)(46 48)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 8 8 8 8 2 ··· 2 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D8 C4○D4 C8⋊C22 kernel C2×C4⋊D8 C2×D4⋊C4 C2×C4⋊C8 C4⋊D8 C2×C4×D4 C2×C4⋊1D4 C22×D8 C42 C22×C4 C2×D4 C2×C4 C2×C4 C22 # reps 1 2 1 8 1 1 2 2 2 4 8 4 2

Matrix representation of C2×C4⋊D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 13 0
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 6 6 0 0 0 0 14 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,6,14,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2×C4⋊D8 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_8
% in TeX

G:=Group("C2xC4:D8");
// GroupNames label

G:=SmallGroup(128,1761);
// by ID

G=gap.SmallGroup(128,1761);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽