Copied to
clipboard

G = C23.35D8order 128 = 27

6th non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.35D8, C24.154D4, C23.32SD16, D43(C22⋊C4), C4.52C22≀C2, (C2×D4).263D4, (C22×D4)⋊17C4, (C22×C8)⋊2C22, (D4×C23).4C2, C22.30(C2×D8), C2.1(C22⋊D8), C23.737(C2×D4), (C22×C4).262D4, C23.7Q83C2, C223(D4⋊C4), C2.1(C22⋊SD16), C22.72C22≀C2, C22.44(C2×SD16), C2.11(C243C4), C22.57(C8⋊C22), (C23×C4).231C22, C23.194(C22⋊C4), (C22×C4).1320C23, (C22×D4).452C22, C2.17(C23.37D4), (C2×C4⋊C4)⋊3C22, C4.1(C2×C22⋊C4), (C2×D4⋊C4)⋊1C2, (C2×C22⋊C8)⋊11C2, (C2×D4).200(C2×C4), C2.17(C2×D4⋊C4), (C2×C4).1310(C2×D4), (C22×C4).260(C2×C4), (C2×C4).358(C22×C4), (C2×C4).119(C22⋊C4), C22.239(C2×C22⋊C4), SmallGroup(128,518)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.35D8
C1C2C22C23C22×C4C23×C4D4×C23 — C23.35D8
C1C2C2×C4 — C23.35D8
C1C23C23×C4 — C23.35D8
C1C2C2C22×C4 — C23.35D8

Generators and relations for C23.35D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 1052 in 424 conjugacy classes, 84 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×4], C4 [×4], C22 [×3], C22 [×8], C22 [×76], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×8], D4 [×28], C23, C23 [×6], C23 [×88], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×12], C2×D4 [×50], C24, C24 [×22], C2.C42, C22⋊C8 [×2], D4⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4 [×2], C22×C8 [×2], C23×C4, C22×D4 [×6], C22×D4 [×11], C25, C23.7Q8, C2×C22⋊C8, C2×D4⋊C4 [×4], D4×C23, C23.35D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×6], D4⋊C4 [×4], C2×C22⋊C4 [×3], C22≀C2 [×4], C2×D8, C2×SD16, C8⋊C22 [×2], C243C4, C2×D4⋊C4, C23.37D4, C22⋊D8 [×2], C22⋊SD16 [×2], C23.35D8

Smallest permutation representation of C23.35D8
On 32 points
Generators in S32
(1 14)(2 28)(3 16)(4 30)(5 10)(6 32)(7 12)(8 26)(9 17)(11 19)(13 21)(15 23)(18 31)(20 25)(22 27)(24 29)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(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 14 23)(2 22 15 27)(3 26 16 21)(4 20 9 25)(5 32 10 19)(6 18 11 31)(7 30 12 17)(8 24 13 29)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A4B4C4D4E4F4G4H4I4J8A···8H
order12···222222···244444444448···8
size11···122224···422224488884···4

38 irreducible representations

dim111111222224
type++++++++++
imageC1C2C2C2C2C4D4D4D4D8SD16C8⋊C22
kernelC23.35D8C23.7Q8C2×C22⋊C8C2×D4⋊C4D4×C23C22×D4C22×C4C2×D4C24C23C23C22
# reps111418381442

Matrix representation of C23.35D8 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
0000516
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
000010
000001
,
3140000
330000
000200
009000
00001116
000016
,
330000
3140000
000200
008000
000061
00001611

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,5,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,9,0,0,0,0,2,0,0,0,0,0,0,0,11,1,0,0,0,0,16,6],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,8,0,0,0,0,2,0,0,0,0,0,0,0,6,16,0,0,0,0,1,11] >;

C23.35D8 in GAP, Magma, Sage, TeX

C_2^3._{35}D_8
% in TeX

G:=Group("C2^3.35D8");
// GroupNames label

G:=SmallGroup(128,518);
// by ID

G=gap.SmallGroup(128,518);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248]);
// Polycyclic

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

׿
×
𝔽