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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×M5(2)
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4×C8 — C4×M5(2)
 Lower central C1 — C2 — C4×M5(2)
 Upper central C1 — C4×C8 — C4×M5(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, M5(2), C2×C42, C22×C8, C4×C16, C165C4, C2×C4×C8, C2×M5(2), C4×M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, M5(2), C2×C42, C22×C8, C2×C4×C8, C2×M5(2), C4×M5(2)

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 8A ··· 8P 8Q ··· 8X 16A ··· 16AF order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C8 C8 C8 M5(2) kernel C4×M5(2) C4×C16 C16⋊5C4 C2×C4×C8 C2×M5(2) C4×C8 M5(2) C2×C42 C22×C8 C42 C2×C8 C22×C4 C4 # reps 1 2 2 1 2 4 16 2 2 8 16 8 16

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

 4 0 0 0 1 0 0 0 1
,
 13 0 0 0 0 1 0 2 0
,
 1 0 0 0 1 0 0 0 16
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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