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G = C233Q16order 128 = 27

2nd semidirect product of C23 and Q16 acting via Q16/C4=C22

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

Aliases: C233Q16, C24.122D4, C4.42+ 1+4, C2.D85C22, C8.18D43C2, C22⋊Q164C2, (C2×Q16)⋊2C22, C4⋊C4.128C23, (C2×C8).151C23, (C2×C4).387C24, Q8⋊C42C22, (C22×C4).485D4, C23.401(C2×D4), C22.17(C2×Q16), C2.14(C22×Q16), C23.48D45C2, (C2×Q8).127C23, C2.68(C233D4), C22⋊C8.176C22, (C23×C4).567C22, (C22×C8).149C22, C22.647(C22×D4), C22⋊Q8.185C22, C2.49(D8⋊C22), (C22×C4).1065C23, (C22×Q8).313C22, (C2×C4).527(C2×D4), (C2×C22⋊C8).32C2, (C2×C22⋊Q8).58C2, (C2×C4⋊C4).637C22, SmallGroup(128,1921)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C233Q16
C1C2C4C2×C4C22×C4C22×Q8C2×C22⋊Q8 — C233Q16
C1C2C2×C4 — C233Q16
C1C22C23×C4 — C233Q16
C1C2C2C2×C4 — C233Q16

Generators and relations for C233Q16
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 404 in 208 conjugacy classes, 94 normal (14 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×25], Q8 [×12], C23, C23 [×6], C23 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×2], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×Q8 [×4], C2×Q8 [×6], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22⋊Q8 [×4], C22×C8 [×2], C2×Q16 [×4], C23×C4, C22×Q8 [×2], C2×C22⋊C8, C22⋊Q16 [×4], C8.18D4 [×4], C23.48D4 [×4], C2×C22⋊Q8 [×2], C233Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C24, C2×Q16 [×6], C22×D4, 2+ 1+4 [×2], C233D4, C22×Q16, D8⋊C22, C233Q16

Smallest permutation representation of C233Q16
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 29)(26 30)(27 31)(28 32)
(2 31)(4 25)(6 27)(8 29)(10 18)(12 20)(14 22)(16 24)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E4F4G···4N8A···8H
order12222···24444444···48···8
size11112···22222448···84···4

32 irreducible representations

dim11111122244
type++++++++-+
imageC1C2C2C2C2C2D4D4Q162+ 1+4D8⋊C22
kernelC233Q16C2×C22⋊C8C22⋊Q16C8.18D4C23.48D4C2×C22⋊Q8C22×C4C24C23C4C2
# reps11444231822

Matrix representation of C233Q16 in GL6(𝔽17)

100000
010000
001000
00161600
0000160
000011
,
1600000
0160000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
3140000
330000
00001615
000001
00161500
000100
,
0130000
1300000
00161500
000100
00001615
000001

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

C233Q16 in GAP, Magma, Sage, TeX

C_2^3\rtimes_3Q_{16}
% in TeX

G:=Group("C2^3:3Q16");
// GroupNames label

G:=SmallGroup(128,1921);
// by ID

G=gap.SmallGroup(128,1921);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,352,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,e*a*e^-1=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^-1=d^-1>;
// generators/relations

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