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G = C22.105C25  order 128 = 27

86th central stem extension by C22 of C25

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

Aliases: C23.52C24, C42.96C23, C22.105C25, C22.22- 1+4, (D4×Q8)⋊22C2, Q8⋊5D4⋊21C2, Q8⋊3Q8⋊21C2, D4⋊3Q8⋊27C2, (C2×C4).95C24, C4⋊C4.498C23, Q8.41(C4○D4), C4⋊Q8.222C22, (C2×D4).477C23, (C4×D4).239C22, C22⋊C4.29C23, (C4×Q8).226C22, (C2×Q8).489C23, Q8○2(C22.D4), C4⋊D4.228C22, (C22×C4).375C23, (C2×C42).954C22, C4.4D4.98C22, C22⋊Q8.120C22, C2.30(C2×2- 1+4), C2.36(C2.C25), C42⋊2C2.18C22, C42.C2.156C22, (C22×Q8).501C22, C22.46C24⋊22C2, C42⋊C2.234C22, C23.32C23⋊16C2, C22.50C24⋊26C2, C22.47C24⋊21C2, C23.36C23⋊36C2, C23.33C23⋊27C2, C22.36C24⋊18C2, C22.35C24⋊13C2, C22.D4.44C22, (C2×C4×Q8)⋊61C2, C4.278(C2×C4○D4), C2.61(C22×C4○D4), (C2×C4⋊C4).711C22, (C2×Q8)○(C22.D4), (C2×C4○D4).233C22, SmallGroup(128,2248)

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

Derived series
C1 — C22 — C22.105C25
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×Q8 — C22.105C25
Lower central
C1 — C22 — C22.105C25
Upper central
C1 — C22 — C22.105C25
Jennings
C1 — C22 — C22.105C25

Generators and relations for C22.105C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=f2=a, g2=b, ab=ba, 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: 660 in 506 conjugacy classes, 390 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, 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.D4, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4×Q8, C23.32C23, C23.33C23, C23.36C23, C22.35C24, C22.36C24, Q8⋊5D4, D4×Q8, C22.46C24, C22.47C24, D4⋊3Q8, C22.50C24, Q8⋊3Q8, C22.105C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.105C25

Smallest permutation representation of C22.105C25
►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 10)(2 9)(3 12)(4 11)(5 30)(6 29)(7 32)(8 31)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 26)(22 25)(23 28)(24 27)(33 45)(34 48)(35 47)(36 46)(37 42)(38 41)(39 44)(40 43)(49 56)(50 55)(51 54)(52 53)

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4P4Q···4AH
order12222222224···44···4
size11112244442···24···4

44 irreducible representations

dim11111111111111244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.105C25C2×C4×Q8C23.32C23C23.33C23C23.36C23C22.35C24C22.36C24Q8⋊5D4D4×Q8C22.46C24C22.47C24D4⋊3Q8C22.50C24Q8⋊3Q8Q8C22C2
# reps11116332133331822

Matrix representation of C22.105C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
400000
110000
000020
000003
003000
000200
,
100000
010000
003000
000300
000020
000002
,
430000
010000
000300
002000
000003
000020
,
400000
040000
000010
000001
004000
000400
,
300000
030000
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,3,1,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

C22.105C25 in GAP, Magma, Sage, TeX 

C_2^2._{105}C_2^5
 
% in TeX
 
G:=Group("C2^2.105C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2248);
 
// by ID
 
G=gap.SmallGroup(128,2248);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,570,136,1684,172]);
 
// 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*b=b*a,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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