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G = C23xSD16  order 128 = 27

Direct product of C23 and SD16

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

Aliases: C23xSD16, C8:3C24, C4.2C25, Q8:1C24, D4.1C24, C24.196D4, (C23xC8):13C2, (C2xC8):16C23, (Q8xC23):13C2, (C2xQ8):19C23, C2.37(D4xC23), C4.28(C22xD4), (C2xC4).608C24, (C22xC8):69C22, (D4xC23).21C2, (C22xC4).628D4, C23.894(C2xD4), (C2xD4).489C23, (C22xQ8):67C22, (C23xC4).712C22, C22.165(C22xD4), (C22xC4).1590C23, (C22xD4).602C22, (C2xC4).881(C2xD4), SmallGroup(128,2307)

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

Derived series
C1 — C4 — C23xSD16
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C23xC4 — D4xC23 — C23xSD16
Lower central
C1 — C2 — C4 — C23xSD16
Upper central
C1 — C24 — C23xC4 — C23xSD16
Jennings
C1 — C2 — C2 — C4 — C23xSD16

Generators and relations for C23xSD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1500 in 860 conjugacy classes, 476 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C24, C22xC8, C2xSD16, C23xC4, C23xC4, C22xD4, C22xD4, C22xQ8, C22xQ8, C25, C23xC8, C22xSD16, D4xC23, Q8xC23, C23xSD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C24, C2xSD16, C22xD4, C25, C22xSD16, D4xC23, C23xSD16

Smallest permutation representation of C23xSD16
►On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2W4A···4H4I···4P8A···8P
order12···22···24···44···48···8
size11···14···42···24···42···2

56 irreducible representations

dim11111222
type+++++++
imageC1C2C2C2C2D4D4SD16
kernelC23xSD16C23xC8C22xSD16D4xC23Q8xC23C22xC4C24C23
# reps1128117116

Matrix representation of C23xSD16 ►in GL5(F17)

10000
016000
001600
00010
00001
,
10000
016000
00100
000160
000016
,
160000
01000
00100
000160
000016
,
10000
01000
001600
00077
00050
,
160000
016000
001600
00010
0001616

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,7,5,0,0,0,7,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16] >;
 

C23xSD16 in GAP, Magma, Sage, TeX 

C_2^3\times {\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3xSD16");
 
// GroupNames label
 
G:=SmallGroup(128,2307);
 
// by ID
 
G=gap.SmallGroup(128,2307);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,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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