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G = C23:3SD16  order 128 = 27

1st semidirect product of C23 and SD16 acting via SD16/C4=C22

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

Aliases: C23:3SD16, C24.80D4, C4:C4:5D4, (C2xC8):19D4, (C2xQ8):5D4, (C2xD4).85D4, C4.61C22wrC2, C4.27(C4:1D4), C2.16(C8:D4), C2.16(C8:8D4), C4.14(C4:D4), C23.897(C2xD4), (C22xC4).138D4, C2.28(D4:D4), C22.4Q16:42C2, C2.18(Q8:D4), (C22xSD16):11C2, C2.5(C23:2D4), C22.90(C2xSD16), C22.193C22wrC2, C2.27(D4.7D4), C2.18(C22:SD16), C22.101(C4oD8), (C23xC4).268C22, (C22xC8).317C22, (C22xD4).57C22, (C22xQ8).46C22, C22.218(C4:D4), C22.128(C8:C22), (C22xC4).1431C23, C22.116(C8.C22), (C2xC22:Q8):1C2, (C2xC22:C8):31C2, (C2xD4:C4):11C2, (C2xQ8:C4):11C2, (C2xC4:D4).12C2, (C2xC4).1021(C2xD4), (C2xC4).614(C4oD4), (C2xC4:C4).102C22, SmallGroup(128,732)

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

Derived series
C1 — C22xC4 — C23:3SD16
Chief series
C1 — C2 — C22 — C23 — C22xC4 — C23xC4 — C2xC4:D4 — C23:3SD16
Lower central
C1 — C2 — C22xC4 — C23:3SD16
Upper central
C1 — C23 — C23xC4 — C23:3SD16
Jennings
C1 — C2 — C2 — C22xC4 — C23:3SD16

Generators and relations for C23:3SD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 528 in 222 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C24, C22:C8, D4:C4, Q8:C4, C2xC22:C4, C2xC4:C4, C2xC4:C4, C4:D4, C22:Q8, C22xC8, C2xSD16, C23xC4, C22xD4, C22xD4, C22xQ8, C22.4Q16, C2xC22:C8, C2xD4:C4, C2xQ8:C4, C2xC4:D4, C2xC22:Q8, C22xSD16, C23:3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C4oD4, C22wrC2, C4:D4, C4:1D4, C2xSD16, C4oD8, C8:C22, C8.C22, C23:2D4, Q8:D4, D4:D4, C22:SD16, D4.7D4, C8:8D4, C8:D4, C23:3SD16

Smallest permutation representation of C23:3SD16
►On 64 points
Generators in S64
(1 52)(2 60)(3 54)(4 62)(5 56)(6 64)(7 50)(8 58)(9 20)(10 27)(11 22)(12 29)(13 24)(14 31)(15 18)(16 25)(17 43)(19 45)(21 47)(23 41)(26 46)(28 48)(30 42)(32 44)(33 63)(34 49)(35 57)(36 51)(37 59)(38 53)(39 61)(40 55)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)

(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(2 4)(3 7)(6 8)(9 54)(10 49)(11 52)(12 55)(13 50)(14 53)(15 56)(16 51)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(34 36)(35 39)(38 40)(41 62)(42 57)(43 60)(44 63)(45 58)(46 61)(47 64)(48 59)
 
G:=sub<Sym(64)| (1,52)(2,60)(3,54)(4,62)(5,56)(6,64)(7,50)(8,58)(9,20)(10,27)(11,22)(12,29)(13,24)(14,31)(15,18)(16,25)(17,43)(19,45)(21,47)(23,41)(26,46)(28,48)(30,42)(32,44)(33,63)(34,49)(35,57)(36,51)(37,59)(38,53)(39,61)(40,55), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,4)(3,7)(6,8)(9,54)(10,49)(11,52)(12,55)(13,50)(14,53)(15,56)(16,51)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(34,36)(35,39)(38,40)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59)>;
 
G:=Group( (1,52)(2,60)(3,54)(4,62)(5,56)(6,64)(7,50)(8,58)(9,20)(10,27)(11,22)(12,29)(13,24)(14,31)(15,18)(16,25)(17,43)(19,45)(21,47)(23,41)(26,46)(28,48)(30,42)(32,44)(33,63)(34,49)(35,57)(36,51)(37,59)(38,53)(39,61)(40,55), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,4)(3,7)(6,8)(9,54)(10,49)(11,52)(12,55)(13,50)(14,53)(15,56)(16,51)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(34,36)(35,39)(38,40)(41,62)(42,57)(43,60)(44,63)(45,58)(46,61)(47,64)(48,59) );
 
G=PermutationGroup([[(1,52),(2,60),(3,54),(4,62),(5,56),(6,64),(7,50),(8,58),(9,20),(10,27),(11,22),(12,29),(13,24),(14,31),(15,18),(16,25),(17,43),(19,45),(21,47),(23,41),(26,46),(28,48),(30,42),(32,44),(33,63),(34,49),(35,57),(36,51),(37,59),(38,53),(39,61),(40,55)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,49),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,4),(3,7),(6,8),(9,54),(10,49),(11,52),(12,55),(13,50),(14,53),(15,56),(16,51),(17,23),(19,21),(20,24),(25,27),(26,30),(29,31),(34,36),(35,39),(38,40),(41,62),(42,57),(43,60),(44,63),(45,58),(46,61),(47,64),(48,59)]])
 

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12···222224444444···48···8
size11···144882222448···84···4

32 irreducible representations

dim1111111122222222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D4C4oD4SD16C4oD8C8:C22C8.C22
kernelC23:3SD16C22.4Q16C2xC22:C8C2xD4:C4C2xQ8:C4C2xC4:D4C2xC22:Q8C22xSD16C4:C4C2xC8C22xC4C2xD4C2xQ8C24C2xC4C23C22C22C22
# reps1111111142122124411

Matrix representation of C23:3SD16 ►in GL6(F17)

040000
1300000
00161500
000100
000001
000010
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1250000
12120000
004800
0001300
000010
0000016
,
100000
0160000
001000
00161600
000010
0000016

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,4,0,0,0,0,0,8,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C23:3SD16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes_3{\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3:3SD16");
 
// GroupNames label
 
G:=SmallGroup(128,732);
 
// by ID
 
G=gap.SmallGroup(128,732);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,248]);
 
// 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,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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