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G = C22.133C25  order 128 = 27

114th central stem extension by C22 of C25

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

Aliases: C23.74C24, C22.133C25, C42.116C23, C4.1112- 1+4, C22.162+ 1+4, Q82⋊12C2, D4⋊6D4⋊37C2, D4⋊3Q8⋊35C2, C4⋊C4.504C23, (C2×C4).123C24, C4⋊Q8.351C22, (C4×D4).247C22, (C2×D4).325C23, C22⋊C4.48C23, (C4×Q8).233C22, (C2×Q8).305C23, C4⋊D4.118C22, C42⋊2C2.5C22, (C2×C42).965C22, (C22×C4).393C23, C22⋊Q8.123C22, C2.43(C2×2- 1+4), C2.62(C2×2+ 1+4), C22.57C24⋊8C2, C4.4D4.102C22, C42.C2.162C22, (C22×Q8).369C22, C22.49C24⋊21C2, C42⋊C2.241C22, C23.36C23⋊47C2, C22.46C24⋊33C2, C23.38C23⋊30C2, C23.41C23⋊20C2, C22.36C24⋊30C2, C22.D4.13C22, (C2×C4⋊Q8)⋊60C2, (C2×C4⋊C4).719C22, (C2×C4○D4).239C22, SmallGroup(128,2276)

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

Derived series
C1 — C22 — C22.133C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊Q8 — C22.133C25
Lower central
C1 — C22 — C22.133C25
Upper central
C1 — C22 — C22.133C25
Jennings
C1 — C22 — C22.133C25

Generators and relations for C22.133C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=a, e2=b, ab=ba, dcd-1=gcg=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 502 conjugacy classes, 384 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, 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, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C23.36C23, C2×C4⋊Q8, C23.38C23, C22.36C24, C23.41C23, D4⋊6D4, C22.46C24, D4⋊3Q8, C22.49C24, Q82, C22.57C24, C22.133C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C22.133C25

Smallest permutation representation of C22.133C25
►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)

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4AB
order122222222244444···4
size111122444422224···4

38 irreducible representations

dim11111111111144
type++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C22- 1+42+ 1+4
kernelC22.133C25C23.36C23C2×C4⋊Q8C23.38C23C22.36C24C23.41C23D4⋊6D4C22.46C24D4⋊3Q8C22.49C24Q82C22.57C24C4C22
# reps12144444211442

Matrix representation of C22.133C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
04000000
00400000
00010000
00001000
00000400
00000040
00000001
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
40000000
04000000
00400000
00040000
00000200
00002000
00000003
00000030
,
20000000
02000000
00300000
00030000
00000200
00002000
00000003
00000030
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C22.133C25 in GAP, Magma, Sage, TeX 

C_2^2._{133}C_2^5
 
% in TeX
 
G:=Group("C2^2.133C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2276);
 
// by ID
 
G=gap.SmallGroup(128,2276);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,352,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=f^2=a,e^2=b,a*b=b*a,d*c*d^-1=g*c*g=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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