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G = C23×C8order 64 = 26

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

direct product, p-group, abelian, monomial

Aliases: C23×C8, SmallGroup(64,246)

Series: Derived Chief Lower central Upper central Jennings

C1 — C23×C8
C1C2C4C2×C4C22×C4C23×C4 — C23×C8
C1 — C23×C8
C1 — C23×C8
C1C2C2C4 — 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 [×14], C4, C4 [×7], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, C23×C8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, C23×C8

Smallest permutation representation of C23×C8
Regular action on 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(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 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(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 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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)| (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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( (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17)(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), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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([(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17),(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)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(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),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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···2O4A···4P8A···8AF
order12···24···48···8
size11···11···11···1

64 irreducible representations

dim111111
type+++
imageC1C2C2C4C4C8
kernelC23×C8C22×C8C23×C4C22×C4C24C23
# reps114114232

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

1000
01600
00160
00016
,
1000
01600
00160
0001
,
1000
01600
0010
00016
,
8000
01500
0040
00013
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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