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## G = C23×C8order 64 = 26

### Abelian group of type [2,2,2,8]

Aliases: C23×C8, SmallGroup(64,246)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23×C8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8
 Lower central C1 — C23×C8
 Upper central C1 — C23×C8
 Jennings C1 — C2 — C2 — C4 — C23×C8

Generators and relations for C23×C8
G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 169, all normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C23×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, C23×C8

Smallest permutation representation of C23×C8
Regular action on 64 points
Generators in S64
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(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)

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

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

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

C23×C8 is a maximal subgroup of
C23.29C42  C24.132D4  C23.36C42  C24.133D4  C23.22D8  C24.19Q8  C23.21M4(2)  C23.22M4(2)  C24.135D4  C23.23D8  C24.5C8  C42.264C23  C24.144D4
C23×C8 is a maximal quotient of
C42.691C23  C42.695C23  C42.697C23  Q8○M5(2)

64 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4P 8A ··· 8AF order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 1 1 ··· 1 1 ··· 1 1 ··· 1

64 irreducible representations

 dim 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C8 kernel C23×C8 C22×C8 C23×C4 C22×C4 C24 C23 # reps 1 14 1 14 2 32

Matrix representation of C23×C8 in GL4(𝔽17) generated by

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

C23×C8 in GAP, Magma, Sage, TeX

C_2^3\times C_8
% in TeX

G:=Group("C2^3xC8");
// GroupNames label

G:=SmallGroup(64,246);
// by ID

G=gap.SmallGroup(64,246);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,96,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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