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G = C24×C4order 64 = 26

Abelian group of type [2,2,2,2,4]

direct product, p-group, abelian, monomial

Aliases: C24×C4, SmallGroup(64,260)

Series: Derived Chief Lower central Upper central Jennings

C1 — C24×C4
C1C2C22C23C24C25 — C24×C4
C1 — C24×C4
C1 — C24×C4
C1C2 — C24×C4

Generators and relations for C24×C4
 G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 681, all normal (4 characteristic)
C1, C2, C2 [×30], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, C24×C4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, C24×C4

Smallest permutation representation of C24×C4
Regular action on 64 points
Generators in S64
(1 47)(2 48)(3 45)(4 46)(5 14)(6 15)(7 16)(8 13)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 32)(25 35)(26 36)(27 33)(28 34)(37 41)(38 42)(39 43)(40 44)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 43)(2 44)(3 41)(4 42)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(37 45)(38 46)(39 47)(40 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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)(3,45)(4,46)(5,14)(6,15)(7,16)(8,13)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32)(25,35)(26,36)(27,33)(28,34)(37,41)(38,42)(39,43)(40,44)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,43)(2,44)(3,41)(4,42)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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)(3,45)(4,46)(5,14)(6,15)(7,16)(8,13)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32)(25,35)(26,36)(27,33)(28,34)(37,41)(38,42)(39,43)(40,44)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,43)(2,44)(3,41)(4,42)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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),(3,45),(4,46),(5,14),(6,15),(7,16),(8,13),(9,17),(10,18),(11,19),(12,20),(21,29),(22,30),(23,31),(24,32),(25,35),(26,36),(27,33),(28,34),(37,41),(38,42),(39,43),(40,44),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,43),(2,44),(3,41),(4,42),(5,18),(6,19),(7,20),(8,17),(9,13),(10,14),(11,15),(12,16),(21,25),(22,26),(23,27),(24,28),(29,35),(30,36),(31,33),(32,34),(37,45),(38,46),(39,47),(40,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,11),(2,12),(3,9),(4,10),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(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)])

C24×C4 is a maximal subgroup of   C24.17Q8  C243C8  C25.85C22  C2413D4  C248Q8  C24.166D4
C24×C4 is a maximal quotient of   C22.14C25  C4.22C25

64 conjugacy classes

class 1 2A···2AE4A···4AF
order12···24···4
size11···11···1

64 irreducible representations

dim1111
type+++
imageC1C2C2C4
kernelC24×C4C23×C4C25C24
# reps130132

Matrix representation of C24×C4 in GL5(𝔽5)

40000
04000
00400
00010
00004
,
40000
01000
00400
00040
00004
,
10000
04000
00100
00040
00004
,
10000
04000
00400
00010
00004
,
40000
04000
00200
00040
00003

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,3] >;

C24×C4 in GAP, Magma, Sage, TeX

C_2^4\times C_4
% in TeX

G:=Group("C2^4xC4");
// GroupNames label

G:=SmallGroup(64,260);
// by ID

G=gap.SmallGroup(64,260);
# by ID

G:=PCGroup([6,-2,2,2,2,2,-2,192]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^4=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,d*e=e*d>;
// generators/relations

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