Copied to
clipboard

G = C2.(C4×D8)  order 128 = 27

3rd central stem extension by C2 of C4×D8

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

Aliases: C2.6(C4×D8), D44(C4⋊C4), C4.53(C4×D4), D4⋊C43C4, C4⋊C4.304D4, (C2×C4).106D8, (C2×D4).27Q8, (C2×D4).273D4, C22.37(C2×D8), C2.2(D4.Q8), C2.3(C22⋊D8), C22.4Q168C2, C2.2(D4⋊Q8), C23.760(C2×D4), (C22×C4).683D4, C22.144(C4×D4), C4.25(C22⋊Q8), C22.84C22≀C2, C2.3(D4.7D4), C22.49(C4○D8), C2.8(SD16⋊C4), C22.70(C8⋊C22), (C2×C42).268C22, (C22×C8).102C22, C4.6(C22.D4), C22.71(C22⋊Q8), (C22×C4).1353C23, C23.65C231C2, C22.7C4215C2, (C22×D4).458C22, C22.59(C8.C22), C2.18(C23.8Q8), C4⋊C44(C2×C4), (C2×C8)⋊5(C2×C4), C4.12(C2×C4⋊C4), (C2×C4×D4).17C2, (C2×C2.D8)⋊4C2, (C2×C4).988(C2×D4), (C2×C4).264(C2×Q8), (C2×D4).161(C2×C4), (C2×D4⋊C4).20C2, (C2×C4).749(C4○D4), (C2×C4⋊C4).757C22, (C2×C4).371(C22×C4), SmallGroup(128,594)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2.(C4×D8)
C1C2C22C2×C4C22×C4C22×D4C2×C4×D4 — C2.(C4×D8)
C1C2C2×C4 — C2.(C4×D8)
C1C23C2×C42 — C2.(C4×D8)
C1C2C2C22×C4 — C2.(C4×D8)

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

Subgroups: 396 in 181 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×8], C22 [×7], C22 [×16], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×24], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42, D4⋊C4 [×4], D4⋊C4 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×3], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×2], C23×C4, C22×D4, C22.7C42, C22.4Q16, C23.65C23, C2×D4⋊C4 [×2], C2×C2.D8, C2×C4×D4, C2.(C4×D8)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.8Q8, C4×D8, SD16⋊C4, C22⋊D8, D4.7D4, D4⋊Q8, D4.Q8, C2.(C4×D8)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4N4O4P4Q4R8A···8H
order12···222224···44···444448···8
size11···144442···24···488884···4

38 irreducible representations

dim11111111222222244
type++++++++++-++-
imageC1C2C2C2C2C2C2C4D4D4D4Q8D8C4○D4C4○D8C8⋊C22C8.C22
kernelC2.(C4×D8)C22.7C42C22.4Q16C23.65C23C2×D4⋊C4C2×C2.D8C2×C4×D4D4⋊C4C4⋊C4C22×C4C2×D4C2×D4C2×C4C2×C4C22C22C22
# reps11112118222244411

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

1600000
0160000
001000
000100
000010
000001
,
1300000
240000
004000
000400
000040
000004
,
16130000
010000
007800
00151000
000033
0000143
,
100000
8160000
001000
00111600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,2,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,7,15,0,0,0,0,8,10,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[1,8,0,0,0,0,0,16,0,0,0,0,0,0,1,11,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2.(C4×D8) in GAP, Magma, Sage, TeX

C_2.(C_4\times D_8)
% in TeX

G:=Group("C2.(C4xD8)");
// GroupNames label

G:=SmallGroup(128,594);
// by ID

G=gap.SmallGroup(128,594);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

׿
×
𝔽