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G = C233SD16order 128 = 27

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

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

Aliases: C233SD16, C24.80D4, C4⋊C45D4, (C2×C8)⋊19D4, (C2×Q8)⋊5D4, (C2×D4).85D4, C4.61C22≀C2, C4.27(C41D4), C2.16(C8⋊D4), C2.16(C88D4), C4.14(C4⋊D4), C23.897(C2×D4), (C22×C4).138D4, C2.28(D4⋊D4), C22.4Q1642C2, C2.18(Q8⋊D4), (C22×SD16)⋊11C2, C2.5(C232D4), C22.90(C2×SD16), C22.193C22≀C2, C2.27(D4.7D4), C2.18(C22⋊SD16), C22.101(C4○D8), (C23×C4).268C22, (C22×C8).317C22, (C22×D4).57C22, (C22×Q8).46C22, C22.218(C4⋊D4), C22.128(C8⋊C22), (C22×C4).1431C23, C22.116(C8.C22), (C2×C22⋊Q8)⋊1C2, (C2×C22⋊C8)⋊31C2, (C2×D4⋊C4)⋊11C2, (C2×Q8⋊C4)⋊11C2, (C2×C4⋊D4).12C2, (C2×C4).1021(C2×D4), (C2×C4).614(C4○D4), (C2×C4⋊C4).102C22, SmallGroup(128,732)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C233SD16
C1C2C22C23C22×C4C23×C4C2×C4⋊D4 — C233SD16
C1C2C22×C4 — C233SD16
C1C23C23×C4 — C233SD16
C1C2C2C22×C4 — C233SD16

Generators and relations for C233SD16
 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 [×7], C2 [×4], C4 [×4], C4 [×7], C22 [×7], C22 [×20], C8 [×3], C2×C4 [×6], C2×C4 [×19], D4 [×14], Q8 [×6], C23, C23 [×2], C23 [×14], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×5], SD16 [×8], C22×C4 [×2], C22×C4 [×9], C2×D4 [×2], C2×D4 [×13], C2×Q8 [×2], C2×Q8 [×5], C24, C24, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C22⋊Q8 [×4], C22×C8 [×2], C2×SD16 [×6], C23×C4, C22×D4, C22×D4, C22×Q8, C22.4Q16, C2×C22⋊C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4⋊D4, C2×C22⋊Q8, C22×SD16, C233SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, SD16 [×2], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C232D4, Q8⋊D4, D4⋊D4, C22⋊SD16, D4.7D4, C88D4, C8⋊D4, C233SD16

Smallest permutation representation of C233SD16
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+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D4C4○D4SD16C4○D8C8⋊C22C8.C22
kernelC233SD16C22.4Q16C2×C22⋊C8C2×D4⋊C4C2×Q8⋊C4C2×C4⋊D4C2×C22⋊Q8C22×SD16C4⋊C4C2×C8C22×C4C2×D4C2×Q8C24C2×C4C23C22C22C22
# reps1111111142122124411

Matrix representation of C233SD16 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] >;

C233SD16 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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