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G = C233D8order 128 = 27

2nd semidirect product of C23 and D8 acting via D8/C4=C22

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

Aliases: C233D8, C24.119D4, C4.12+ 1+4, C87D43C2, C22⋊D84C2, (C2×D8)⋊2C22, C2.D83C22, (C22×C8)⋊9C22, C22.23(C2×D8), C2.14(C22×D8), D4⋊C41C22, C4⋊C4.125C23, C4⋊D456C22, C22⋊C853C22, (C2×C4).384C24, (C2×C8).150C23, (C22×C4).482D4, C23.399(C2×D4), C22.D84C2, (C2×D4).137C23, (C22×D4)⋊23C22, C2.65(C233D4), (C23×C4).564C22, C22.644(C22×D4), C2.47(D8⋊C22), (C22×C4).1062C23, (C2×C4⋊D4)⋊49C2, (C2×C4⋊C4)⋊50C22, (C2×C22⋊C8)⋊22C2, (C2×C4).525(C2×D4), SmallGroup(128,1918)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C233D8
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C233D8
C1C2C2×C4 — C233D8
C1C22C23×C4 — C233D8
C1C2C2C2×C4 — C233D8

Generators and relations for C233D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 660 in 260 conjugacy classes, 94 normal (14 characteristic)
C1, C2 [×3], C2 [×10], C4 [×2], C4 [×7], C22, C22 [×6], C22 [×30], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×28], C23, C23 [×6], C23 [×18], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], D8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×4], C2×D4 [×22], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×C22⋊C8, C22⋊D8 [×4], C87D4 [×4], C22.D8 [×4], C2×C4⋊D4 [×2], C233D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, 2+ 1+4 [×2], C233D4, C22×D8, D8⋊C22, C233D8

Smallest permutation representation of C233D8
On 32 points
Generators in S32
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 5)(2 26)(3 7)(4 28)(6 30)(8 32)(9 23)(10 14)(11 17)(12 16)(13 19)(15 21)(18 22)(20 24)(25 29)(27 31)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 2)(3 8)(4 7)(5 6)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 26)(27 32)(28 31)(29 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J8A···8H
order12222···2222244444444448···8
size11112···2888822224488884···4

32 irreducible representations

dim11111122244
type++++++++++
imageC1C2C2C2C2C2D4D4D82+ 1+4D8⋊C22
kernelC233D8C2×C22⋊C8C22⋊D8C87D4C22.D8C2×C4⋊D4C22×C4C24C23C4C2
# reps11444231822

Matrix representation of C233D8 in GL6(𝔽17)

1600000
0160000
0001300
004000
000004
0000130
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
3140000
330000
000001
000010
001000
0001600
,
330000
3140000
000010
000001
001000
000100

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

C233D8 in GAP, Magma, Sage, TeX

C_2^3\rtimes_3D_8
% in TeX

G:=Group("C2^3:3D8");
// GroupNames label

G:=SmallGroup(128,1918);
// by ID

G=gap.SmallGroup(128,1918);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,4037,1027,124]);
// Polycyclic

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

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