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## G = C23×D8order 128 = 27

### Direct product of C23 and D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C23×D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C23×D8
 Lower central C1 — C2 — C4 — C23×D8
 Upper central C1 — C24 — C23×C4 — C23×D8
 Jennings C1 — C2 — C2 — C4 — C23×D8

Generators and relations for C23×D8
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=d-1 >

Subgroups: 2012 in 988 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, D4, C23, C23, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C24, C22×C8, C2×D8, C23×C4, C22×D4, C22×D4, C25, C23×C8, C22×D8, D4×C23, C23×D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C25, C22×D8, D4×C23, C23×D8

Smallest permutation representation of C23×D8
On 64 points
Generators in S64
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(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)
(1 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 4A ··· 4H 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D4 D8 kernel C23×D8 C23×C8 C22×D8 D4×C23 C22×C4 C24 C23 # reps 1 1 28 2 7 1 16

Matrix representation of C23×D8 in GL5(𝔽17)

 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 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 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
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 3 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 16 1

G:=sub<GL(5,GF(17))| [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],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,11,11],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,0,1] >;

C23×D8 in GAP, Magma, Sage, TeX

C_2^3\times D_8
% in TeX

G:=Group("C2^3xD8");
// GroupNames label

G:=SmallGroup(128,2306);
// by ID

G=gap.SmallGroup(128,2306);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,4037,2028,124]);
// 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^-1>;
// generators/relations

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