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## G = C22×M5(2)  order 128 = 27

### Direct product of C22 and M5(2)

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×M5(2)
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C23×C8 — C22×M5(2)
 Lower central C1 — C2 — C22×M5(2)
 Upper central C1 — C22×C8 — C22×M5(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C22×M5(2)

Generators and relations for C22×M5(2)
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 220 in 200 conjugacy classes, 180 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C23, C23, C23, C16, C2×C8, C22×C4, C22×C4, C24, C2×C16, M5(2), C22×C8, C22×C8, C23×C4, C22×C16, C2×M5(2), C23×C8, C22×M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, M5(2), C22×C8, C23×C4, C2×M5(2), C23×C8, C22×M5(2)

Smallest permutation representation of C22×M5(2)
On 64 points
Generators in S64
(1 62)(2 63)(3 64)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 48)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 49)
(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 30)(2 23)(3 32)(4 25)(5 18)(6 27)(7 20)(8 29)(9 22)(10 31)(11 24)(12 17)(13 26)(14 19)(15 28)(16 21)(33 50)(34 59)(35 52)(36 61)(37 54)(38 63)(39 56)(40 49)(41 58)(42 51)(43 60)(44 53)(45 62)(46 55)(47 64)(48 57)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 8A ··· 8P 8Q ··· 8X 16A ··· 16AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 type + + + + image C1 C2 C2 C2 C4 C4 C8 C8 M5(2) kernel C22×M5(2) C22×C16 C2×M5(2) C23×C8 C22×C8 C23×C4 C22×C4 C24 C22 # reps 1 2 12 1 14 2 28 4 16

Matrix representation of C22×M5(2) in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 13 0 0 0 0 13 0 0 0 0 0 1 0 0 8 0
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,8,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;

C22×M5(2) in GAP, Magma, Sage, TeX

C_2^2\times M_5(2)
% in TeX

G:=Group("C2^2xM5(2)");
// GroupNames label

G:=SmallGroup(128,2137);
// by ID

G=gap.SmallGroup(128,2137);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,925,102,124]);
// Polycyclic

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

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