Copied to
clipboard

G = C42.357C23order 128 = 27

218th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.357C23, C4⋊Q89C22, C4⋊C816C22, (C4×C8)⋊41C22, C4⋊C4.350D4, D4.Q822C2, C8⋊C47C22, (C4×SD16)⋊30C2, D4.D48C2, C4⋊C4.76C23, (C2×C8).50C23, (C4×Q8)⋊10C22, C2.D826C22, C4.Q868C22, D4.13(C4○D4), (C2×C4).321C24, C22⋊C4.151D4, C22⋊SD16.2C2, (C4×D4).83C22, C23.260(C2×D4), SD16⋊C416C2, (C2×Q8).82C23, C42.C23C22, C2.30(D4○SD16), Q8⋊C426C22, (C2×D4).411C23, C22⋊C8.34C22, C22⋊Q8.29C22, D4⋊C4.37C22, C23.20D421C2, (C22×C4).294C23, C42.7C226C2, C22.11C24.9C2, C22.581(C22×D4), C22.35C242C2, (C2×SD16).145C22, (C22×D4).363C22, C42⋊C2.132C22, C2.122(C22.19C24), C4.206(C2×C4○D4), (C2×C4).505(C2×D4), SmallGroup(128,1855)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.357C23
C1C2C4C2×C4C42C4×D4C22.11C24 — C42.357C23
C1C2C2×C4 — C42.357C23
C1C22C42⋊C2 — C42.357C23
C1C2C2C2×C4 — C42.357C23

Generators and relations for C42.357C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b2, ab=ba, ac=ca, dad=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd=a2b2c, ece=bc, de=ed >

Subgroups: 396 in 197 conjugacy classes, 88 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×4], D4 [×6], Q8 [×4], C23, C23 [×8], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×8], C2×C8 [×4], SD16 [×8], C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], C4⋊Q8, C2×SD16 [×4], C22×D4, C42.7C22, C4×SD16 [×2], SD16⋊C4 [×2], C22⋊SD16 [×2], D4.D4 [×2], D4.Q8 [×2], C23.20D4 [×2], C22.11C24, C22.35C24, C42.357C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○SD16 [×2], C42.357C23

Smallest permutation representation of C42.357C23
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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 30 15 12)(6 31 16 9)(7 32 13 10)(8 29 14 11)
(1 10 20 32)(2 11 17 29)(3 12 18 30)(4 9 19 31)(5 26 15 21)(6 27 16 22)(7 28 13 23)(8 25 14 24)
(2 17)(4 19)(5 13)(6 8)(7 15)(9 11)(10 30)(12 32)(14 16)(22 27)(24 25)(29 31)
(1 18)(2 19)(3 20)(4 17)(5 10)(6 11)(7 12)(8 9)(13 30)(14 31)(15 32)(16 29)(21 23)(22 24)(25 27)(26 28)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2H4A···4F4G···4M4N4O4P4Q8A8B8C8D8E8F
order12222···24···44···44444888888
size11114···42···24···48888444488

32 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16
kernelC42.357C23C42.7C22C4×SD16SD16⋊C4C22⋊SD16D4.D4D4.Q8C23.20D4C22.11C24C22.35C24C22⋊C4C4⋊C4D4C2
# reps11222222112284

Matrix representation of C42.357C23 in GL6(𝔽17)

400000
040000
0040150
0004015
00160130
00016013
,
100000
010000
000100
0016000
000001
0000160
,
120000
0160000
00143107
003377
00512314
0012121414
,
100000
16160000
001000
000100
0040160
0004016
,
100000
010000
0016000
000100
0000160
000001

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,16,0,0,0,0,4,0,16,0,0,15,0,13,0,0,0,0,15,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,14,3,5,12,0,0,3,3,12,12,0,0,10,7,3,14,0,0,7,7,14,14],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,4,0,0,0,0,1,0,4,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C42.357C23 in GAP, Magma, Sage, TeX

C_4^2._{357}C_2^3
% in TeX

G:=Group("C4^2.357C2^3");
// GroupNames label

G:=SmallGroup(128,1855);
// by ID

G=gap.SmallGroup(128,1855);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,1018,304,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽