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G = C22.139C25  order 128 = 27

120th central stem extension by C22 of C25

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

Aliases: C23.80C24, C22.139C25, C42.122C23, C4.452- 1+4, C4.952+ 1+4, Q82⋊13C2, (D4×Q8)⋊29C2, C4⋊C4.506C23, (C2×C4).129C24, C4⋊Q8.226C22, (C2×D4).331C23, (C4×D4).250C22, C22⋊C4.54C23, (C2×Q8).309C23, (C4×Q8).236C22, C4⋊1D4.192C22, C4⋊D4.234C22, (C22×C4).399C23, (C2×C42).968C22, C42⋊2C2.6C22, C22⋊Q8.236C22, C2.68(C2×2+ 1+4), C2.46(C2×2- 1+4), C4.4D4.104C22, C42.C2.164C22, (C22×Q8).371C22, C22.53C24⋊22C2, C42⋊C2.244C22, C22.50C24⋊34C2, C23.37C23⋊51C2, C22.36C24⋊34C2, C23.38C23⋊33C2, C22.57C24⋊10C2, C22.26C24.51C2, C22.D4.16C22, (C2×C4○D4).243C22, SmallGroup(128,2282)

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

Derived series
C1 — C22 — C22.139C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.139C25
Lower central
C1 — C22 — C22.139C25
Upper central
C1 — C22 — C22.139C25
Jennings
C1 — C22 — C22.139C25

Generators and relations for C22.139C25
 G = < a,b,c,d,e,f,g | a2=b2=1, c2=d2=f2=g2=a, e2=b, ab=ba, dcd-1=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 700 in 501 conjugacy classes, 384 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C2×C42, C42⋊C2, C4×D4, 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, C22.26C24, C23.37C23, C23.38C23, C22.36C24, D4×Q8, C22.50C24, Q82, C22.53C24, C22.57C24, C22.139C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C22.139C25

Smallest permutation representation of C22.139C25
►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 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 43 3 41)(2 42 4 44)(5 59 7 57)(6 58 8 60)(9 37 11 39)(10 40 12 38)(13 36 15 34)(14 35 16 33)(17 29 19 31)(18 32 20 30)(21 55 23 53)(22 54 24 56)(25 51 27 49)(26 50 28 52)(45 63 47 61)(46 62 48 64)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2H4A···4F4G···4AC
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim111111111144
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C22+ 1+42- 1+4
kernelC22.139C25C22.26C24C23.37C23C23.38C23C22.36C24D4×Q8C22.50C24Q82C22.53C24C22.57C24C4C4
# reps112484422424

Matrix representation of C22.139C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
00020000
00200000
02000000
20000000
00002000
00000300
00000020
00000003
,
00040000
00100000
04000000
10000000
00000004
00000010
00000400
00001000
,
00100000
00010000
40000000
04000000
00004000
00000400
00000040
00000004
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
01000000
40000000
00010000
00400000
00000100
00004000
00000001
00000040

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3],[0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0],[0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

C22.139C25 in GAP, Magma, Sage, TeX 

C_2^2._{139}C_2^5
 
% in TeX
 
G:=Group("C2^2.139C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2282);
 
// by ID
 
G=gap.SmallGroup(128,2282);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,723,352,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=1,c^2=d^2=f^2=g^2=a,e^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^-1=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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