Copied to
clipboard

?

G = C2×C8⋊7D4  order 128 = 27

Direct product of C2 and C8⋊7D4

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

Aliases: C2×C8⋊7D4, C23⋊4D8, C24.142D4, C8⋊16(C2×D4), (C2×C8)⋊37D4, (C23×C8)⋊8C2, C22⋊1(C2×D8), C2.6(C22×D8), (C2×D8)⋊42C22, (C22×D8)⋊10C2, C4⋊C4.17C23, C2.D8⋊44C22, C4⋊D4⋊53C22, (C2×C4).252C24, (C2×C8).486C23, (C22×C8)⋊65C22, (C2×D4).57C23, (C22×C4).606D4, C4.146(C22×D4), C23.859(C2×D4), C4.108(C4⋊D4), D4⋊C4⋊57C22, C22.91(C4○D8), (C23×C4).701C22, C22.512(C22×D4), C22.172(C4⋊D4), (C22×C4).1531C23, (C22×D4).345C22, (C2×C2.D8)⋊18C2, C4.19(C2×C4○D4), C2.14(C2×C4○D8), (C2×C4⋊D4)⋊46C2, C2.70(C2×C4⋊D4), (C2×D4⋊C4)⋊17C2, (C2×C4).1424(C2×D4), (C2×C4).698(C4○D4), (C2×C4⋊C4).586C22, SmallGroup(128,1780)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C2×C8⋊7D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C8⋊7D4
Lower central
C1 — C2 — C2×C4 — C2×C8⋊7D4
Upper central
C1 — C23 — C23×C4 — C2×C8⋊7D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C8⋊7D4

Subgroups: 692 in 308 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×28], C23, C23 [×6], C23 [×20], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×10], D8 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×26], C24, C24 [×2], D4⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×D8 [×4], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×D4⋊C4 [×2], C2×C2.D8, C8⋊7D4 [×8], C2×C4⋊D4 [×2], C23×C8, C22×D8, C2×C8⋊7D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×D8 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C8⋊7D4 [×4], C2×C4⋊D4, C22×D8, C2×C4○D8, C2×C8⋊7D4

Generators and relations
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1430000
14140000
0014300
00141400
0000160
0000016
,
1600000
010000
0016000
000100
0000162
0000161
,
100000
0160000
0016000
000100
000010
0000116

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],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L8A···8P
order12···2222222224···444448···8
size11···1222288882···288882···2

44 irreducible representations

dim1111111222222
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4C4○D4D8C4○D8
kernelC2×C8⋊7D4C2×D4⋊C4C2×C2.D8C8⋊7D4C2×C4⋊D4C23×C8C22×D8C2×C8C22×C4C24C2×C4C23C22
# reps1218211431488

In GAP, Magma, Sage, TeX 

C_2\times C_8\rtimes_7D_4
 
% in TeX
 
G:=Group("C2xC8:7D4");
 
// GroupNames label
 
G:=SmallGroup(128,1780);
 
// by ID
 
G=gap.SmallGroup(128,1780);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172]);
 
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀