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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

Aliases: C22×M5(2), C16⋊4C23, C24.6C8, C8.23C24, (C2×C4)○M5(2), C4○(C2×M5(2)), C8○2(C2×M5(2)), (C2×C8)○2M5(2), (C22×C16)⋊14C2, (C2×C16)⋊21C22, (C22×C8).50C4, C23.41(C2×C8), (C23×C8).26C2, C2.10(C23×C8), C4.62(C23×C4), C8.63(C22×C4), (C22×C4).18C8, C4.39(C22×C8), (C23×C4).43C4, (C2×C8).616C23, C22.16(C22×C8), (C22×C8).586C22, (C2×C4).90(C2×C8), (C2×C8)○(C2×M5(2)), (C2×C4)○(C2×M5(2)), (C2×C8).255(C2×C4), (C2×C4).627(C22×C4), (C22×C4).507(C2×C4), SmallGroup(128,2137)

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···2G2H2I2J2K4A···4H4I4J4K4L8A···8P8Q···8X16A···16AF
order12···222224···444448···88···816···16
size11···122221···122221···12···22···2

80 irreducible representations

dim111111112
type++++
imageC1C2C2C2C4C4C8C8M5(2)
kernelC22×M5(2)C22×C16C2×M5(2)C23×C8C22×C8C23×C4C22×C4C24C22
# reps1212114228416

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

1000
01600
00160
00016
,
16000
01600
00160
00016
,
13000
01300
0001
0080
,
1000
0100
00160
0001
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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