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G = C22.145C25  order 128 = 27

126th central stem extension by C22 of C25

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

Aliases: C23.86C24, C42.128C23, C22.145C25, C4.482- 1+4, Q8⋊3Q8⋊28C2, C4⋊C4.330C23, (C2×C4).135C24, C4⋊Q8.228C22, C22⋊C4.59C23, (C4×Q8).241C22, (C2×Q8).471C23, (C2×C42).973C22, C42⋊2C2.8C22, C22⋊Q8.237C22, C2.51(C2×2- 1+4), C42.C2.87C22, (C22×C4).1219C23, C22.58C24⋊3C2, C42⋊C2.249C22, C22.35C24.5C2, C23.37C23.46C2, SmallGroup(128,2288)

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

Derived series
C1 — C22 — C22.145C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.145C25
Lower central
C1 — C22 — C22.145C25
Upper central
C1 — C22 — C22.145C25
Jennings
C1 — C22 — C22.145C25

Generators and relations for C22.145C25
 G = < a,b,c,d,e,f,g | a2=b2=1, c2=f2=g2=a, d2=ba=ab, e2=b, 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: 508 in 445 conjugacy classes, 384 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C23.37C23, C22.35C24, Q8⋊3Q8, C22.58C24, C22.145C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C22.145C25

Smallest permutation representation of C22.145C25
►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 57 49 15)(2 60 50 14)(3 59 51 13)(4 58 52 16)(5 41 34 27)(6 44 35 26)(7 43 36 25)(8 42 33 28)(9 63 55 17)(10 62 56 20)(11 61 53 19)(12 64 54 18)(21 47 39 29)(22 46 40 32)(23 45 37 31)(24 48 38 30)

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

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

(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), (1,57,49,15)(2,60,50,14)(3,59,51,13)(4,58,52,16)(5,41,34,27)(6,44,35,26)(7,43,36,25)(8,42,33,28)(9,63,55,17)(10,62,56,20)(11,61,53,19)(12,64,54,18)(21,47,39,29)(22,46,40,32)(23,45,37,31)(24,48,38,30), (1,55,51,11)(2,12,52,56)(3,53,49,9)(4,10,50,54)(5,29,36,45)(6,46,33,30)(7,31,34,47)(8,48,35,32)(13,17,57,61)(14,62,58,18)(15,19,59,63)(16,64,60,20)(21,41,37,25)(22,26,38,42)(23,43,39,27)(24,28,40,44), (1,39,3,37)(2,24,4,22)(5,17,7,19)(6,62,8,64)(9,41,11,43)(10,26,12,28)(13,47,15,45)(14,32,16,30)(18,35,20,33)(21,51,23,49)(25,55,27,53)(29,57,31,59)(34,63,36,61)(38,52,40,50)(42,56,44,54)(46,58,48,60), (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), (1,57,49,15)(2,60,50,14)(3,59,51,13)(4,58,52,16)(5,41,34,27)(6,44,35,26)(7,43,36,25)(8,42,33,28)(9,63,55,17)(10,62,56,20)(11,61,53,19)(12,64,54,18)(21,47,39,29)(22,46,40,32)(23,45,37,31)(24,48,38,30), (1,55,51,11)(2,12,52,56)(3,53,49,9)(4,10,50,54)(5,29,36,45)(6,46,33,30)(7,31,34,47)(8,48,35,32)(13,17,57,61)(14,62,58,18)(15,19,59,63)(16,64,60,20)(21,41,37,25)(22,26,38,42)(23,43,39,27)(24,28,40,44), (1,39,3,37)(2,24,4,22)(5,17,7,19)(6,62,8,64)(9,41,11,43)(10,26,12,28)(13,47,15,45)(14,32,16,30)(18,35,20,33)(21,51,23,49)(25,55,27,53)(29,57,31,59)(34,63,36,61)(38,52,40,50)(42,56,44,54)(46,58,48,60), (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)], [(1,57,49,15),(2,60,50,14),(3,59,51,13),(4,58,52,16),(5,41,34,27),(6,44,35,26),(7,43,36,25),(8,42,33,28),(9,63,55,17),(10,62,56,20),(11,61,53,19),(12,64,54,18),(21,47,39,29),(22,46,40,32),(23,45,37,31),(24,48,38,30)], [(1,55,51,11),(2,12,52,56),(3,53,49,9),(4,10,50,54),(5,29,36,45),(6,46,33,30),(7,31,34,47),(8,48,35,32),(13,17,57,61),(14,62,58,18),(15,19,59,63),(16,64,60,20),(21,41,37,25),(22,26,38,42),(23,43,39,27),(24,28,40,44)], [(1,39,3,37),(2,24,4,22),(5,17,7,19),(6,62,8,64),(9,41,11,43),(10,26,12,28),(13,47,15,45),(14,32,16,30),(18,35,20,33),(21,51,23,49),(25,55,27,53),(29,57,31,59),(34,63,36,61),(38,52,40,50),(42,56,44,54),(46,58,48,60)], [(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 2A2B2C2D4A···4F4G···4AG
order122224···44···4
size111142···24···4

38 irreducible representations

dim111114
type+++++-
imageC1C2C2C2C22- 1+4
kernelC22.145C25C23.37C23C22.35C24Q8⋊3Q8C22.58C24C4
# reps13121246

Matrix representation of C22.145C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01300000
10030000
00040000
00400000
00000003
00000030
00000300
00003000
,
01000000
40000000
40010000
01400000
00000010
00000001
00004000
00000400
,
20000000
03000000
00300000
00020000
00004000
00000400
00000040
00000004
,
10000000
01000000
01400000
10040000
00000400
00001000
00000001
00000040
,
10000000
01000000
00100000
00010000
00000100
00004000
00000001
00000040

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

C22.145C25 in GAP, Magma, Sage, TeX 

C_2^2._{145}C_2^5
 
% in TeX
 
G:=Group("C2^2.145C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2288);
 
// by ID
 
G=gap.SmallGroup(128,2288);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,680,1430,723,352,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=1,c^2=f^2=g^2=a,d^2=b*a=a*b,e^2=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=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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