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G = C4×M5(2)  order 128 = 27

Direct product of C4 and M5(2)

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

Aliases: C4×M5(2), C82M5(2), C42.10C8, C8.17C42, C169(C2×C4), C4.5(C4×C8), (C4×C16)⋊14C2, (C4×C8).37C4, (C2×C8).18C8, C8.25(C2×C8), (C2×C8)M5(2), C8(C2×M5(2)), C42(C165C4), C82(C165C4), C165C414C2, C22.5(C4×C8), (C22×C8).46C4, (C2×C42).46C4, (C22×C4).10C8, C4.36(C2×C42), (C2×C4).73C42, C23.34(C2×C8), C8.66(C22×C4), C4.35(C22×C8), C2.2(C2×M5(2)), C42(C2×M5(2)), C42(C165C4), (C2×C8).621C23, C42.340(C2×C4), (C4×C8).442C22, (C2×C16).104C22, (C2×M5(2)).27C2, C22.24(C22×C8), (C22×C8).575C22, (C2×C4×C8).65C2, C2.11(C2×C4×C8), (C2×C4).99(C2×C8), (C4×C8)(C2×M5(2)), (C4×C8)(C165C4), (C2×C8).192(C2×C4), (C2×C4).606(C22×C4), (C22×C4).484(C2×C4), SmallGroup(128,839)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×M5(2)
C1C2C4C2×C4C2×C8C22×C8C2×C4×C8 — C4×M5(2)
C1C2 — C4×M5(2)
C1C4×C8 — C4×M5(2)
C1C2C2C2C2C4C4C2×C8 — C4×M5(2)

Generators and relations for C4×M5(2)
 G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Subgroups: 108 in 98 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C16 [×8], C42 [×2], C42 [×2], C2×C8 [×4], C2×C8 [×8], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C16 [×4], M5(2) [×8], C2×C42, C22×C8 [×2], C4×C16 [×2], C165C4 [×2], C2×C4×C8, C2×M5(2) [×2], C4×M5(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], M5(2) [×4], C2×C42, C22×C8 [×2], C2×C4×C8, C2×M5(2) [×2], C4×M5(2)

Smallest permutation representation of C4×M5(2)
On 64 points
Generators in S64
(1 34 51 32)(2 35 52 17)(3 36 53 18)(4 37 54 19)(5 38 55 20)(6 39 56 21)(7 40 57 22)(8 41 58 23)(9 42 59 24)(10 43 60 25)(11 44 61 26)(12 45 62 27)(13 46 63 28)(14 47 64 29)(15 48 49 30)(16 33 50 31)
(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 51)(2 60)(3 53)(4 62)(5 55)(6 64)(7 57)(8 50)(9 59)(10 52)(11 61)(12 54)(13 63)(14 56)(15 49)(16 58)(17 43)(18 36)(19 45)(20 38)(21 47)(22 40)(23 33)(24 42)(25 35)(26 44)(27 37)(28 46)(29 39)(30 48)(31 41)(32 34)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8X16A···16AF
order1222224···44···48···88···816···16
size1111221···12···21···12···22···2

80 irreducible representations

dim1111111111112
type+++++
imageC1C2C2C2C2C4C4C4C4C8C8C8M5(2)
kernelC4×M5(2)C4×C16C165C4C2×C4×C8C2×M5(2)C4×C8M5(2)C2×C42C22×C8C42C2×C8C22×C4C4
# reps1221241622816816

Matrix representation of C4×M5(2) in GL3(𝔽17) generated by

400
010
001
,
1300
001
020
,
100
010
0016
G:=sub<GL(3,GF(17))| [4,0,0,0,1,0,0,0,1],[13,0,0,0,0,2,0,1,0],[1,0,0,0,1,0,0,0,16] >;

C4×M5(2) in GAP, Magma, Sage, TeX

C_4\times M_5(2)
% in TeX

G:=Group("C4xM5(2)");
// GroupNames label

G:=SmallGroup(128,839);
// by ID

G=gap.SmallGroup(128,839);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,1430,136,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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