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G = C234SD16order 128 = 27

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

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

Aliases: C234SD16, C24.120D4, C4.22+ 1+4, C88D425C2, Q8⋊D428C2, C4.Q831C22, C22⋊SD1628C2, C4⋊C4.126C23, C22⋊C868C22, (C2×C8).319C23, (C2×C4).385C24, (C22×C8)⋊36C22, C23.400(C2×D4), (C22×C4).483D4, C22⋊Q868C22, D4⋊C442C22, Q8⋊C446C22, (C2×SD16)⋊38C22, (C2×D4).138C23, (C2×Q8).125C23, C22.37(C2×SD16), C2.21(C22×SD16), (C22×Q8)⋊19C22, C23.46D428C2, C23.47D428C2, C4⋊D4.180C22, C2.66(C233D4), (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

C1C2×C4 — C234SD16
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C234SD16
C1C2C2×C4 — C234SD16
C1C22C23×C4 — C234SD16
C1C2C2C2×C4 — C234SD16

Generators and relations for C234SD16
 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 [×3], C2 [×8], C4 [×2], C4 [×9], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×6], C23 [×3], C23 [×4], C23 [×10], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], SD16 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C2×Q8 [×3], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×C22⋊C8, Q8⋊D4 [×2], C22⋊SD16 [×2], C88D4 [×4], C23.46D4 [×2], C23.47D4 [×2], C2×C4⋊D4, C2×C22⋊Q8, C234SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C22×D4, 2+ 1+4 [×2], C233D4, C22×SD16, D8⋊C22, C234SD16

Smallest permutation representation of C234SD16
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
kernelC234SD16C2×C22⋊C8Q8⋊D4C22⋊SD16C88D4C23.46D4C23.47D4C2×C4⋊D4C2×C22⋊Q8C22×C4C24C23C4C2
# reps11224221131822

Matrix representation of C234SD16 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] >;

C234SD16 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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