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G = C22.111C25  order 128 = 27

92nd central stem extension by C22 of C25

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

Aliases: C23.55C24, C22.111C25, C42.102C23, C4.422- 1+4, C22.202+ 1+4, Q82⋊11C2, Q8○3(C22⋊Q8), D4⋊6D4⋊30C2, Q8⋊14(C4○D4), Q8⋊5D4⋊24C2, D4⋊3Q8⋊31C2, Q8⋊6D4⋊23C2, C4⋊C4.502C23, (C2×C4).101C24, C4⋊Q8.349C22, (C4×D4).242C22, (C2×D4).483C23, C22⋊C4.34C23, (C4×Q8).229C22, (C2×Q8).491C23, C4⋊D4.230C22, C4⋊1D4.189C22, (C2×C42).958C22, (C22×C4).380C23, C22⋊Q8.121C22, C2.44(C2×2+ 1+4), C2.33(C2×2- 1+4), C22.26C24⋊46C2, C42⋊2C2.21C22, C4.4D4.100C22, C42.C2.157C22, (C22×Q8).502C22, C22.53C24⋊17C2, C42⋊C2.237C22, C23.33C23⋊31C2, C22.36C24⋊22C2, C23.36C23⋊42C2, C22.D4.33C22, (C2×C4×Q8)⋊62C2, C4.284(C2×C4○D4), C2.67(C22×C4○D4), (C2×C4⋊C4).713C22, (C2×C4○D4).235C22, SmallGroup(128,2254)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22.111C25
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×Q8 — C22.111C25
Lower central
C1 — C22 — C22.111C25
Upper central
C1 — C22 — C22.111C25
Jennings
C1 — C22 — C22.111C25

Generators and relations for C22.111C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=f2=a, g2=ba=ab, dcd-1=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 796 in 552 conjugacy classes, 392 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4×Q8, C23.33C23, C23.36C23, C22.26C24, C22.36C24, D4⋊6D4, Q8⋊5D4, Q8⋊6D4, D4⋊3Q8, Q82, C22.53C24, C22.111C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C25, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C22.111C25

Smallest permutation representation of C22.111C25
►On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)

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

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4P4Q···4AF
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim111111111111244
type++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D42- 1+42+ 1+4
kernelC22.111C25C2×C4×Q8C23.33C23C23.36C23C22.26C24C22.36C24D4⋊6D4Q8⋊5D4Q8⋊6D4D4⋊3Q8Q82C22.53C24Q8C4C22
# reps112336621313822

Matrix representation of C22.111C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
020000
300000
001000
000400
000010
000004
,
100000
010000
000400
001000
000001
000040
,
010000
100000
001000
000100
000010
000001
,
100000
010000
000010
000001
004000
000400
,
200000
020000
000100
004000
000001
000040

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

C22.111C25 in GAP, Magma, Sage, TeX 

C_2^2._{111}C_2^5
 
% in TeX
 
G:=Group("C2^2.111C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2254);
 
// by ID
 
G=gap.SmallGroup(128,2254);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,570,136,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=f^2=a,g^2=b*a=a*b,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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