Copied to
clipboard

G = 2+ 1+45C4order 128 = 27

4th semidirect product of 2+ 1+4 and C4 acting via C4/C2=C2

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

Aliases: 2+ 1+45C4, D4(D4⋊C4), Q8(Q8⋊C4), C4○D4.50D4, C2.1(D4○D8), C4.8(C23×C4), (C2×D4).343D4, (C2×Q8).262D4, C4⋊C4.344C23, (C2×C4).178C24, (C2×C8).468C23, (C22×C8)⋊49C22, C2.1(D4○SD16), D4.20(C22×C4), C23.431(C2×D4), C4.143(C22×D4), Q8.20(C22×C4), D4⋊C486C22, D4.19(C22⋊C4), C42⋊C24C22, Q8⋊C489C22, (C2×D4).362C23, Q8.19(C22⋊C4), (C2×Q8).335C23, C23.37D431C2, C23.36D436C2, C23.24D435C2, (C2×M4(2))⋊71C22, (C22×C4).902C23, C22.128(C22×D4), (C2×2+ 1+4).6C2, C23.33C232C2, (C22×D4).320C22, C4○D41(C2×C4), (C2×C8○D4)⋊17C2, (C2×D4)⋊24(C2×C4), (C2×C4⋊C4)⋊43C22, (C2×Q8)(Q8⋊C4), (C2×C4).446(C2×D4), C4.33(C2×C22⋊C4), (C2×D4⋊C4)⋊50C2, (C2×C4).62(C22×C4), C22.6(C2×C22⋊C4), (C2×C4○D4).84C22, C2.40(C22×C22⋊C4), SmallGroup(128,1629)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — 2+ 1+45C4
C1C2C22C2×C4C22×C4C2×C4○D4C2×2+ 1+4 — 2+ 1+45C4
C1C2C4 — 2+ 1+45C4
C1C22C2×C4○D4 — 2+ 1+45C4
C1C2C2C2×C4 — 2+ 1+45C4

Generators and relations for 2+ 1+45C4
 G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, be=eb, dcd=ece-1=a2c, ede-1=cd >

Subgroups: 764 in 389 conjugacy classes, 172 normal (18 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×30], C8 [×4], C2×C4, C2×C4 [×15], C2×C4 [×20], D4 [×18], D4 [×27], Q8 [×6], Q8, C23 [×3], C23 [×21], C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×3], C4⋊C4 [×3], C2×C8, C2×C8 [×3], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×18], C2×D4 [×36], C2×Q8 [×2], C4○D4 [×20], C4○D4 [×14], C24 [×3], D4⋊C4 [×12], Q8⋊C4, Q8⋊C4 [×3], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×3], C4×Q8, C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×4], C22×D4 [×3], C22×D4 [×3], C2×C4○D4, C2×C4○D4 [×3], C2×C4○D4, 2+ 1+4 [×8], 2+ 1+4 [×4], C2×D4⋊C4 [×3], C23.24D4 [×3], C23.36D4 [×3], C23.37D4 [×3], C23.33C23, C2×C8○D4, C2×2+ 1+4, 2+ 1+45C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, D4○D8, D4○SD16, 2+ 1+45C4

Smallest permutation representation of 2+ 1+45C4
On 32 points
Generators in S32
(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)
(1 23)(2 22)(3 21)(4 24)(5 11)(6 10)(7 9)(8 12)(13 26)(14 25)(15 28)(16 27)(17 31)(18 30)(19 29)(20 32)
(1 9 3 11)(2 10 4 12)(5 23 7 21)(6 24 8 22)(13 32 15 30)(14 29 16 31)(17 25 19 27)(18 26 20 28)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 26 22 14)(2 25 23 13)(3 28 24 16)(4 27 21 15)(5 32 12 19)(6 31 9 18)(7 30 10 17)(8 29 11 20)

G:=sub<Sym(32)| (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), (1,23)(2,22)(3,21)(4,24)(5,11)(6,10)(7,9)(8,12)(13,26)(14,25)(15,28)(16,27)(17,31)(18,30)(19,29)(20,32), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,32,15,30)(14,29,16,31)(17,25,19,27)(18,26,20,28), (1,22)(2,23)(3,24)(4,21)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,26,22,14)(2,25,23,13)(3,28,24,16)(4,27,21,15)(5,32,12,19)(6,31,9,18)(7,30,10,17)(8,29,11,20)>;

G:=Group( (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), (1,23)(2,22)(3,21)(4,24)(5,11)(6,10)(7,9)(8,12)(13,26)(14,25)(15,28)(16,27)(17,31)(18,30)(19,29)(20,32), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,32,15,30)(14,29,16,31)(17,25,19,27)(18,26,20,28), (1,22)(2,23)(3,24)(4,21)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,26,22,14)(2,25,23,13)(3,28,24,16)(4,27,21,15)(5,32,12,19)(6,31,9,18)(7,30,10,17)(8,29,11,20) );

G=PermutationGroup([(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)], [(1,23),(2,22),(3,21),(4,24),(5,11),(6,10),(7,9),(8,12),(13,26),(14,25),(15,28),(16,27),(17,31),(18,30),(19,29),(20,32)], [(1,9,3,11),(2,10,4,12),(5,23,7,21),(6,24,8,22),(13,32,15,30),(14,29,16,31),(17,25,19,27),(18,26,20,28)], [(1,22),(2,23),(3,24),(4,21),(5,10),(6,11),(7,12),(8,9),(13,19),(14,20),(15,17),(16,18),(25,32),(26,29),(27,30),(28,31)], [(1,26,22,14),(2,25,23,13),(3,28,24,16),(4,27,21,15),(5,32,12,19),(6,31,9,18),(7,30,10,17),(8,29,11,20)])

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A···4H4I···4R8A8B8C8D8E···8J
order12222···22···24···44···488888···8
size11112···24···42···24···422224···4

44 irreducible representations

dim11111111122244
type++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4D4D4○D8D4○SD16
kernel2+ 1+45C4C2×D4⋊C4C23.24D4C23.36D4C23.37D4C23.33C23C2×C8○D4C2×2+ 1+42+ 1+4C2×D4C2×Q8C4○D4C2C2
# reps133331111631422

Matrix representation of 2+ 1+45C4 in GL6(𝔽17)

100000
010000
0000016
000010
0001600
001000
,
100000
010000
000001
0000160
0001600
001000
,
1600000
0160000
000100
0016000
000001
0000160
,
1600000
310000
001000
0001600
0000160
000001
,
11130000
560000
0000314
00001414
0031400
00141400

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[11,5,0,0,0,0,13,6,0,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,3,14,0,0,0,0,14,14,0,0] >;

2+ 1+45C4 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_5C_4
% in TeX

G:=Group("ES+(2,2):5C4");
// GroupNames label

G:=SmallGroup(128,1629);
// by ID

G=gap.SmallGroup(128,1629);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172]);
// Polycyclic

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

׿
×
𝔽