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G = C22.148C25  order 128 = 27

129th central stem extension by C22 of C25

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

Aliases: C23.89C24, C22.148C25, C42.585C23, C4.1142- 1+4, C22.242+ 1+4, D4⋊6D4⋊43C2, D4⋊3Q8⋊43C2, C4⋊C4.508C23, (C2×C4).138C24, C4⋊Q8.356C22, (C4×D4).256C22, (C2×D4).337C23, (C4×Q8).243C22, (C2×Q8).314C23, C4⋊D4.123C22, C4⋊1D4.194C22, C22⋊C4.118C23, (C2×C42).975C22, (C22×C4).407C23, C22⋊Q8.130C22, C2.73(C2×2+ 1+4), C2.53(C2×2- 1+4), C2.59(C2.C25), C22.26C24⋊55C2, C42⋊2C2.27C22, C4.4D4.183C22, C42.C2.168C22, C42⋊C2.251C22, C23.36C23⋊57C2, C22.47C24⋊37C2, C22.33C24⋊21C2, C23.41C23⋊24C2, C22.56C24⋊14C2, C22.57C24⋊16C2, C22.31C24⋊28C2, C22.34C24⋊26C2, C22.D4.21C22, (C2×C42.C2)⋊51C2, (C2×C4⋊C4).727C22, (C2×C4○D4).246C22, SmallGroup(128,2291)

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

Derived series
C1 — C22 — C22.148C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42.C2 — C22.148C25
Lower central
C1 — C22 — C22.148C25
Upper central
C1 — C22 — C22.148C25
Jennings
C1 — C22 — C22.148C25

Generators and relations for C22.148C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=a, f2=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: 756 in 512 conjugacy classes, 382 normal (38 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, C4○D4, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×C4○D4, C2×C42.C2, C23.36C23, C22.26C24, C22.31C24, C22.33C24, C22.34C24, C23.41C23, D4⋊6D4, C22.47C24, D4⋊3Q8, C22.56C24, C22.57C24, C22.148C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C2.C25, C22.148C25

Smallest permutation representation of C22.148C25
►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 46)(2 45)(3 48)(4 47)(5 54)(6 53)(7 56)(8 55)(9 33)(10 36)(11 35)(12 34)(13 38)(14 37)(15 40)(16 39)(17 28)(18 27)(19 26)(20 25)(21 58)(22 57)(23 60)(24 59)(29 52)(30 51)(31 50)(32 49)(41 64)(42 63)(43 62)(44 61)

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

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

(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), (1,46)(2,45)(3,48)(4,47)(5,54)(6,53)(7,56)(8,55)(9,33)(10,36)(11,35)(12,34)(13,38)(14,37)(15,40)(16,39)(17,28)(18,27)(19,26)(20,25)(21,58)(22,57)(23,60)(24,59)(29,52)(30,51)(31,50)(32,49)(41,64)(42,63)(43,62)(44,61), (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), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,55,51,11)(2,54,52,10)(3,53,49,9)(4,56,50,12)(5,45,36,29)(6,48,33,32)(7,47,34,31)(8,46,35,30)(13,61,57,17)(14,64,58,20)(15,63,59,19)(16,62,60,18)(21,41,37,25)(22,44,38,28)(23,43,39,27)(24,42,40,26), (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), (1,46)(2,45)(3,48)(4,47)(5,54)(6,53)(7,56)(8,55)(9,33)(10,36)(11,35)(12,34)(13,38)(14,37)(15,40)(16,39)(17,28)(18,27)(19,26)(20,25)(21,58)(22,57)(23,60)(24,59)(29,52)(30,51)(31,50)(32,49)(41,64)(42,63)(43,62)(44,61), (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), (1,13,3,15)(2,58,4,60)(5,41,7,43)(6,26,8,28)(9,19,11,17)(10,64,12,62)(14,50,16,52)(18,54,20,56)(21,31,23,29)(22,48,24,46)(25,34,27,36)(30,38,32,40)(33,42,35,44)(37,47,39,45)(49,59,51,57)(53,63,55,61), (1,55,51,11)(2,54,52,10)(3,53,49,9)(4,56,50,12)(5,45,36,29)(6,48,33,32)(7,47,34,31)(8,46,35,30)(13,61,57,17)(14,64,58,20)(15,63,59,19)(16,62,60,18)(21,41,37,25)(22,44,38,28)(23,43,39,27)(24,42,40,26), (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)], [(1,46),(2,45),(3,48),(4,47),(5,54),(6,53),(7,56),(8,55),(9,33),(10,36),(11,35),(12,34),(13,38),(14,37),(15,40),(16,39),(17,28),(18,27),(19,26),(20,25),(21,58),(22,57),(23,60),(24,59),(29,52),(30,51),(31,50),(32,49),(41,64),(42,63),(43,62),(44,61)], [(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)], [(1,13,3,15),(2,58,4,60),(5,41,7,43),(6,26,8,28),(9,19,11,17),(10,64,12,62),(14,50,16,52),(18,54,20,56),(21,31,23,29),(22,48,24,46),(25,34,27,36),(30,38,32,40),(33,42,35,44),(37,47,39,45),(49,59,51,57),(53,63,55,61)], [(1,55,51,11),(2,54,52,10),(3,53,49,9),(4,56,50,12),(5,45,36,29),(6,48,33,32),(7,47,34,31),(8,46,35,30),(13,61,57,17),(14,64,58,20),(15,63,59,19),(16,62,60,18),(21,41,37,25),(22,44,38,28),(23,43,39,27),(24,42,40,26)], [(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 2A2B2C2D2E2F···2K4A4B4C4D4E···4Z
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim1111111111111444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C22- 1+42+ 1+4C2.C25
kernelC22.148C25C2×C42.C2C23.36C23C22.26C24C22.31C24C22.33C24C22.34C24C23.41C23D4⋊6D4C22.47C24D4⋊3Q8C22.56C24C22.57C24C4C22C2
# reps1111262264222222

Matrix representation of C22.148C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01300000
40020000
40010000
01400000
00001000
00000400
00000010
00001014
,
01300000
10030000
00040000
00400000
00000200
00002000
00003234
00003202
,
40000000
04000000
04100000
40010000
00003000
00000300
00000030
00000003
,
01000000
40000000
40010000
01400000
00000010
00001413
00001000
00000001
,
40000000
04000000
00400000
00040000
00000100
00001000
00001413
00000004

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

C22.148C25 in GAP, Magma, Sage, TeX 

C_2^2._{148}C_2^5
 
% in TeX
 
G:=Group("C2^2.148C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2291);
 
// by ID
 
G=gap.SmallGroup(128,2291);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,184,2019,570,248,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=a,f^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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