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G = C2×C4⋊C8order 64 = 26

Direct product of C2 and C4⋊C8

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

Aliases: C2×C4⋊C8, C42.10C4, C42.66C22, C22.10M4(2), C4(C4⋊C8), (C2×C4)⋊3C8, C42(C2×C8), C4.71(C2×D4), C4.21(C4⋊C4), C4.20(C2×Q8), (C2×C4).25Q8, (C2×C4).146D4, C2.2(C22×C8), (C22×C8).6C2, C23.39(C2×C4), (C2×C42).13C2, (C22×C4).14C4, C22.10(C2×C8), (C2×C8).60C22, C2.4(C2×M4(2)), C22.18(C4⋊C4), (C2×C4).147C23, C22.21(C22×C4), (C22×C4).136C22, (C2×C4)(C4⋊C8), C2.3(C2×C4⋊C4), (C2×C4).85(C2×C4), SmallGroup(64,103)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C4⋊C8
C1C2C4C2×C4C22×C4C2×C42 — C2×C4⋊C8
C1C2 — C2×C4⋊C8
C1C22×C4 — C2×C4⋊C8
C1C2C2C2×C4 — C2×C4⋊C8

Generators and relations for C2×C4⋊C8
 G = < a,b,c | a2=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 81 in 69 conjugacy classes, 57 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×12], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×3], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C2×C4⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8

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

C2×C4⋊C8 is a maximal subgroup of
C42.385D4  C42.46Q8  C42.3Q8  C42.25D4  C42.27D4  C42.8Q8  C42.389D4  C22.M5(2)  C42.397D4  C42.45D4  C42.46D4  C42.47D4  C42.409D4  C42.410D4  C42.78D4  C42.79D4  C42.80D4  C42.81D4  M4(2)⋊1C8  C42.90D4  C42.91D4  C42.92D4  C43.7C2  C42.45Q8  C8×C4⋊C4  C4⋊C813C4  C4⋊C814C4  C42.425D4  C42.95D4  C42.98D4  C42.99D4  C42.100D4  C42.101D4  C428C8  C42.23Q8  C429C8  C42.25Q8  C23.21M4(2)  (C2×C8).195D4  C4⋊C43C8  (C2×C8).Q8  C22⋊C44C8  C23.9M4(2)  C42.61Q8  C42.27Q8  C42.29Q8  C42.30Q8  C42.31Q8  C42.430D4  C42.325D4  C42.109D4  C42.117D4  C42.118D4  C42.119D4  C42.327D4  C42.120D4  C42.121D4  C42.122D4  C42.123D4  (C2×C4)⋊3D8  (C2×C4)⋊5SD16  (C2×C4)⋊3Q16  C4⋊C4.106D4  (C2×Q8).8Q8  (C2×C4).23D8  (C2×C8).52D4  (C2×C4).26D8  (C2×C4).21Q16  C4.(C4⋊Q8)  C42.674C23  C42.678C23  D4×C2×C8  M4(2)⋊23D4  Q8×C2×C8  M4(2)⋊9Q8  C42.697C23  C42.698C23  D48M4(2)  C42.307C23  C42.309C23  C42.443D4  C42.447D4  C42.293D4  C42.294D4  C42.295D4  C42.296D4  C42.297D4  C42.298D4  C42.11F5
C2×C4⋊C8 is a maximal quotient of
C42.42Q8  M4(2)⋊1C8  C42.425D4  C428C8  C429C8  C23.21M4(2)  C42.61Q8  C4⋊M5(2)  C4⋊C4.7C8  M4(2).1C8  C42.11F5

40 conjugacy classes

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

40 irreducible representations

dim1111111222
type+++++-
imageC1C2C2C2C4C4C8D4Q8M4(2)
kernelC2×C4⋊C8C4⋊C8C2×C42C22×C8C42C22×C4C2×C4C2×C4C2×C4C22
# reps14124416224

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

16000
01600
0010
0001
,
16000
01600
00016
0010
,
9000
0400
00150
0002
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[9,0,0,0,0,4,0,0,0,0,15,0,0,0,0,2] >;

C2×C4⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes C_8
% in TeX

G:=Group("C2xC4:C8");
// GroupNames label

G:=SmallGroup(64,103);
// by ID

G=gap.SmallGroup(64,103);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,88]);
// Polycyclic

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

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