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G = C2×C8⋊C4order 64 = 26

Direct product of C2 and C8⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C8⋊C4, C4.8C42, C42.5C4, C22.9C42, C42.58C22, C22.8M4(2), C89(C2×C4), (C2×C8)⋊8C4, C4(C8⋊C4), C2.5(C2×C42), (C22×C4).9C4, (C2×C42).5C2, C23.38(C2×C4), (C22×C8).14C2, (C2×C8).97C22, C4.32(C22×C4), C2.1(C2×M4(2)), (C2×C4).141C23, C22.17(C22×C4), (C22×C4).135C22, (C2×C4)(C8⋊C4), (C2×C4).83(C2×C4), SmallGroup(64,84)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C8⋊C4
C1C2C22C2×C4C22×C4C2×C42 — C2×C8⋊C4
C1C2 — C2×C8⋊C4
C1C22×C4 — C2×C8⋊C4
C1C2C2C2×C4 — C2×C8⋊C4

Generators and relations for C2×C8⋊C4
 G = < a,b,c | a2=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 81 in 73 conjugacy classes, 65 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×4], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], C8⋊C4 [×4], C2×C42, C22×C8 [×2], C2×C8⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C2×C8⋊C4

Smallest permutation representation of C2×C8⋊C4
Regular action on 64 points
Generators in S64
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)
(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 32 59 9)(2 29 60 14)(3 26 61 11)(4 31 62 16)(5 28 63 13)(6 25 64 10)(7 30 57 15)(8 27 58 12)(17 33 55 46)(18 38 56 43)(19 35 49 48)(20 40 50 45)(21 37 51 42)(22 34 52 47)(23 39 53 44)(24 36 54 41)

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

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

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

C2×C8⋊C4 is a maximal subgroup of
C42.20D4  C42.2Q8  C42.7Q8  C42.370D4  C42.2C8  M5(2)⋊C4  C42.66D4  C42.67D4  C42.68D4  C42.69D4  C2.C43  C43.C2  (C4×C8)⋊12C4  C42.378D4  C42.379D4  C23.36C42  C23.17C42  D4⋊C42  Q8⋊C42  D4.3C42  C43.7C2  C42.45Q8  C4⋊C813C4  C4⋊C814C4  C8.5C42  C8⋊C42  C8.6C42  C42.24Q8  C42.104D4  C42.26Q8  C42.106D4  (C2×C8).Q8  C23.9M4(2)  C2.(C8⋊D4)  C2.(C82D4)  C42.107D4  C42.27Q8  C4.(C4×Q8)  C8⋊(C4⋊C4)  C42.28Q8  C42.109D4  C42.110D4  C42.111D4  C42.112D4  (C2×Q16)⋊10C4  (C2×D8)⋊10C4  C8⋊(C22⋊C4)  C42.116D4  C42.120D4  C42.124D4  C42.125D4  C2×C4×M4(2)  D4.5C42  C42.261C23  C42.266C23  C42.383D4  C42.287C23  C42.293C23  D46M4(2)  C42.239D4  C42.247D4  C42.252D4  C42.257D4  C42.258D4  C42.5F5
C2×C8⋊C4 is a maximal quotient of
C89M4(2)  C23.27C42  C424C8  C42.378D4  C23.36C42  C43.7C2  C4⋊C813C4  C8.23C42  C42.5F5

40 conjugacy classes

class 1 2A···2G4A···4H4I···4P8A···8P
order12···24···44···48···8
size11···11···12···22···2

40 irreducible representations

dim11111112
type++++
imageC1C2C2C2C4C4C4M4(2)
kernelC2×C8⋊C4C8⋊C4C2×C42C22×C8C42C2×C8C22×C4C22
# reps141241648

Matrix representation of C2×C8⋊C4 in GL4(𝔽17) generated by

16000
01600
0010
0001
,
16000
0400
001010
00107
,
1000
01300
0001
00160
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,4,0,0,0,0,10,10,0,0,10,7],[1,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0] >;

C2×C8⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes C_4
% in TeX

G:=Group("C2xC8:C4");
// GroupNames label

G:=SmallGroup(64,84);
// by ID

G=gap.SmallGroup(64,84);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,409,103,117]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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