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G = C23⋊4SD16  order 128 = 27

2nd semidirect product of C23 and SD16 acting via SD16/C4=C22

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

Aliases: C23⋊4SD16, C24.120D4, C4.22+ 1+4, C8⋊8D4⋊25C2, Q8⋊D4⋊28C2, C4.Q8⋊31C22, C22⋊SD16⋊28C2, C4⋊C4.126C23, C22⋊C8⋊68C22, (C2×C8).319C23, (C2×C4).385C24, (C22×C8)⋊36C22, C23.400(C2×D4), (C22×C4).483D4, C22⋊Q8⋊68C22, D4⋊C4⋊42C22, Q8⋊C4⋊46C22, (C2×SD16)⋊38C22, (C2×D4).138C23, (C2×Q8).125C23, C22.37(C2×SD16), C2.21(C22×SD16), (C22×Q8)⋊19C22, C23.46D4⋊28C2, C23.47D4⋊28C2, C4⋊D4.180C22, C2.66(C23⋊3D4), (C23×C4).565C22, C22.645(C22×D4), C2.48(D8⋊C22), (C22×C4).1063C23, (C22×D4).380C22, (C2×C22⋊C8)⋊36C2, (C2×C4⋊C4)⋊51C22, (C2×C4).526(C2×D4), (C2×C22⋊Q8)⋊57C2, (C2×C4⋊D4).59C2, SmallGroup(128,1919)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C23⋊4SD16
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4⋊D4 — C23⋊4SD16
Lower central
C1 — C2 — C2×C4 — C23⋊4SD16
Upper central
C1 — C22 — C23×C4 — C23⋊4SD16
Jennings
C1 — C2 — C2 — C2×C4 — C23⋊4SD16

Generators and relations for C23⋊4SD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 532 in 234 conjugacy classes, 94 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×SD16, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C22⋊C8, Q8⋊D4, C22⋊SD16, C8⋊8D4, C23.46D4, C23.47D4, C2×C4⋊D4, C2×C22⋊Q8, C23⋊4SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C22×D4, 2+ 1+4, C23⋊3D4, C22×SD16, D8⋊C22, C23⋊4SD16

Smallest permutation representation of C23⋊4SD16
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12222···2224444444···48···8
size11112···2882222448···84···4

32 irreducible representations

dim11111111122244
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4SD162+ 1+4D8⋊C22
kernelC23⋊4SD16C2×C22⋊C8Q8⋊D4C22⋊SD16C8⋊8D4C23.46D4C23.47D4C2×C4⋊D4C2×C22⋊Q8C22×C4C24C23C4C2
# reps11224221131822

Matrix representation of C23⋊4SD16 ►in GL6(𝔽17)

100000
010000
0016200
000100
0000016
0000160
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
5120000
550000
0000150
0000161
009000
0091600
,
0160000
1600000
0016000
0016100
000010
0000016

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,16,0,0,15,16,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C23⋊4SD16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes_4{\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3:4SD16");
 
// GroupNames label
 
G:=SmallGroup(128,1919);
 
// by ID
 
G=gap.SmallGroup(128,1919);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,675,4037,1027,124]);
 
// 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=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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