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G = C22.113C25  order 128 = 27

94th central stem extension by C22 of C25

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

Aliases: C23.56C24, C42.104C23, C22.113C25, Q8⋊6D4⋊24C2, Q8⋊5D4⋊25C2, Q8⋊3Q8⋊24C2, Q8○(C42⋊2C2), C4⋊C4.503C23, (C2×C4).103C24, Q8.44(C4○D4), C4⋊Q8.224C22, (C4×D4).244C22, (C2×D4).485C23, (C2×Q8).492C23, (C4×Q8).231C22, C4⋊D4.231C22, C4⋊1D4.116C22, C22⋊C4.112C23, (C22×C4).381C23, (C2×C42).960C22, C22⋊Q8.122C22, C2.40(C2.C25), C42⋊2C2.29C22, C4.4D4.178C22, C42.C2.159C22, (C22×Q8).366C22, C23.36C23⋊43C2, C22.53C24⋊18C2, C23.33C23⋊32C2, C22.34C24⋊15C2, C22.46C24⋊26C2, C22.49C24⋊18C2, C22.47C24⋊25C2, C23.32C23⋊19C2, C42⋊C2.239C22, C22.36C24⋊23C2, C22.D4.35C22, (C4×C4○D4)⋊38C2, C4⋊C4○(C42⋊2C2), C4.286(C2×C4○D4), (C2×Q8)○(C42⋊2C2), C2.69(C22×C4○D4), (C2×C4⋊C4).714C22, (C2×C4○D4).236C22, SmallGroup(128,2256)

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

Derived series
C1 — C22 — C22.113C25
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — C22.113C25
Lower central
C1 — C22 — C22.113C25
Upper central
C1 — C22 — C22.113C25
Jennings
C1 — C22 — C22.113C25

Subgroups: 716 in 515 conjugacy classes, 388 normal (36 characteristic)
C1, C2 [×3], C2 [×7], C4 [×6], C4 [×21], C22, C22 [×21], C2×C4 [×6], C2×C4 [×18], C2×C4 [×30], D4 [×27], Q8 [×4], Q8 [×9], C23, C23 [×6], C42, C42 [×21], C22⋊C4 [×3], C22⋊C4 [×39], C4⋊C4 [×3], C4⋊C4 [×39], C22×C4 [×21], C2×D4 [×21], C2×Q8, C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×8], C2×C42 [×3], C2×C4⋊C4 [×3], C42⋊C2 [×18], C4×D4 [×27], C4×Q8 [×4], C4×Q8 [×9], C4⋊D4 [×21], C22⋊Q8 [×9], C22.D4 [×18], C4.4D4 [×15], C42.C2 [×9], C42⋊2C2, C42⋊2C2 [×9], C4⋊1D4 [×3], C4⋊Q8 [×3], C22×Q8, C2×C4○D4 [×2], C4×C4○D4, C23.32C23, C23.33C23, C23.36C23 [×6], C22.34C24 [×3], C22.36C24 [×3], Q8⋊5D4 [×2], Q8⋊6D4, C22.46C24 [×3], C22.47C24 [×3], C22.49C24 [×3], Q8⋊3Q8, C22.53C24 [×3], C22.113C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.113C25

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
100000
000002
000020
000300
003000
,
400000
040000
000030
000003
003000
000300
,
400000
010000
002000
000200
000020
000002
,
100000
010000
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2,0,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,3,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[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],[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] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2J4A···4R4S···4AG
order12222···24···44···4
size11114···42···24···4

44 irreducible representations

dim1111111111111124
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.113C25C4×C4○D4C23.32C23C23.33C23C23.36C23C22.34C24C22.36C24Q8⋊5D4Q8⋊6D4C22.46C24C22.47C24C22.49C24Q8⋊3Q8C22.53C24Q8C2
# reps1111633213331384

In GAP, Magma, Sage, TeX 

C_2^2._{113}C_2^5
 
% in TeX
 
G:=Group("C2^2.113C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2256);
 
// by ID
 
G=gap.SmallGroup(128,2256);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^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^-1=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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