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G = C22.100C25  order 128 = 27

81st central stem extension by C22 of C25

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

Aliases: C23.49C24, C42.92C23, C22.100C25, C4.1082- 1+4, C4.1572+ 1+4, D4⋊6D4⋊24C2, D4⋊3Q8⋊24C2, Q8⋊5D4⋊19C2, (C2×C4).90C24, C4⋊C4.495C23, C4⋊Q8.345C22, (C4×D4).236C22, (C2×D4).307C23, (C2×Q8).292C23, (C4×Q8).224C22, C4⋊D4.227C22, C4⋊1D4.196C22, C22⋊C4.109C23, (C2×C42).951C22, C42⋊2C2.3C22, (C22×C4).370C23, C4.4D4.97C22, C22⋊Q8.118C22, C2.38(C2×2+ 1+4), C2.28(C2×2- 1+4), C22.26C24⋊41C2, C42.C2.154C22, (C22×Q8).363C22, C23.37C23⋊43C2, C22.50C24⋊24C2, C42⋊C2.231C22, C23.33C23⋊23C2, C22.31C24⋊17C2, C22.47C24⋊17C2, C22.36C24⋊15C2, C22.49C24⋊14C2, C22.D4.10C22, C4⋊C4○(C4⋊Q8), C4⋊C4○(C4⋊1D4), (C2×C4⋊Q8)⋊58C2, (C4×C4○D4)⋊31C2, (C2×C4)⋊8(C4○D4), C4.141(C2×C4○D4), C22.36(C2×C4○D4), C2.56(C22×C4○D4), (C2×C4⋊C4).708C22, (C2×C4○D4).230C22, SmallGroup(128,2243)

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

Derived series
C1 — C22 — C22.100C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.100C25
Lower central
C1 — C22 — C22.100C25
Upper central
C1 — C22 — C22.100C25
Jennings
C1 — C22 — C22.100C25

Generators and relations for C22.100C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=b, f2=g2=a, ab=ba, 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 >

Subgroups: 796 in 552 conjugacy classes, 392 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, 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×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C4×C4○D4, C23.33C23, C2×C4⋊Q8, C22.26C24, C22.26C24, C23.37C23, C22.31C24, C22.36C24, D4⋊6D4, Q8⋊5D4, C22.47C24, D4⋊3Q8, C22.49C24, C22.50C24, C22.100C25
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, C22.100C25

Smallest permutation representation of C22.100C25
►On 64 points
Generators in S64
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

(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)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111111111111244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC22.100C25C4×C4○D4C23.33C23C2×C4⋊Q8C22.26C24C23.37C23C22.31C24C22.36C24D4⋊6D4Q8⋊5D4C22.47C24D4⋊3Q8C22.49C24C22.50C24C2×C4C4C4
# reps11213224244222822

Matrix representation of C22.100C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
140000
001000
000400
000010
000004
,
300000
030000
000002
000030
000200
003000
,
130000
140000
004000
000400
000040
000004
,
100000
010000
000010
000001
004000
000400
,
100000
010000
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],[1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

C22.100C25 in GAP, Magma, Sage, TeX 

C_2^2._{100}C_2^5
 
% in TeX
 
G:=Group("C2^2.100C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2243);
 
// by ID
 
G=gap.SmallGroup(128,2243);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=b,f^2=g^2=a,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^-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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