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G = C23⋊3SD16order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C23⋊3SD16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C23⋊3SD16
 Lower central C1 — C2 — C22×C4 — C23⋊3SD16
 Upper central C1 — C23 — C23×C4 — C23⋊3SD16
 Jennings C1 — C2 — C2 — C22×C4 — C23⋊3SD16

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 ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G ··· 4L 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 8 8 2 2 2 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D4 C4○D4 SD16 C4○D8 C8⋊C22 C8.C22 kernel C23⋊3SD16 C22.4Q16 C2×C22⋊C8 C2×D4⋊C4 C2×Q8⋊C4 C2×C4⋊D4 C2×C22⋊Q8 C22×SD16 C4⋊C4 C2×C8 C22×C4 C2×D4 C2×Q8 C24 C2×C4 C23 C22 C22 C22 # reps 1 1 1 1 1 1 1 1 4 2 1 2 2 1 2 4 4 1 1

Matrix representation of C233SD16 in GL6(𝔽17)

 0 4 0 0 0 0 13 0 0 0 0 0 0 0 16 15 0 0 0 0 0 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 5 0 0 0 0 12 12 0 0 0 0 0 0 4 8 0 0 0 0 0 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 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

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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