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G = C22.154C25order 128 = 27

135th central stem extension by C22 of C25

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

Aliases: C23.94C24, C42.136C23, C22.154C25, C4.1162- 1+4, D43Q846C2, Q83Q831C2, C4⋊C4.509C23, (C2×C4).144C24, C4⋊Q8.230C22, (C2×D4).342C23, (C4×D4).258C22, C22⋊C4.64C23, (C4×Q8).245C22, (C2×Q8).319C23, C4⋊D4.125C22, (C2×C42).979C22, (C22×C4).413C23, C22⋊Q8.238C22, C2.57(C2×2- 1+4), C42.C2.89C22, C2.65(C2.C25), C422C2.11C22, C4.4D4.109C22, C23.37C2358C2, C22.56C2417C2, C22.57C2421C2, C23.36C2362C2, C22.33C2425C2, C22.46C2441C2, C23.41C2327C2, C42⋊C2.253C22, C22.47C2440C2, C22.36C2442C2, C22.35C2426C2, C22.50C2441C2, C22.49C2426C2, C22.D4.22C22, (C2×C4⋊C4).729C22, SmallGroup(128,2297)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.154C25
C1C2C22C2×C4C42C2×C42C23.36C23 — C22.154C25
C1C22 — C22.154C25
C1C22 — C22.154C25
C1C22 — C22.154C25

Generators and relations for C22.154C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=1, c2=b, e2=g2=a, f2=ba=ab, dcd=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: 620 in 465 conjugacy classes, 380 normal (122 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×25], C22, C22 [×15], C2×C4 [×12], C2×C4 [×14], C2×C4 [×18], D4 [×11], Q8 [×11], C23 [×3], C23 [×2], C42 [×6], C42 [×14], C22⋊C4 [×10], C22⋊C4 [×30], C4⋊C4 [×16], C4⋊C4 [×44], C22×C4 [×5], C22×C4 [×10], C2×D4 [×4], C2×D4 [×6], C2×Q8 [×2], C2×Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×6], C42⋊C2 [×8], C4×D4 [×6], C4×D4 [×6], C4×Q8 [×4], C4×Q8 [×8], C4⋊D4 [×4], C4⋊D4 [×6], C22⋊Q8 [×4], C22⋊Q8 [×22], C22.D4 [×2], C22.D4 [×14], C4.4D4 [×2], C4.4D4 [×10], C42.C2 [×6], C42.C2 [×12], C422C2 [×2], C422C2 [×18], C4⋊Q8 [×4], C4⋊Q8 [×6], C23.36C23 [×2], C23.37C23, C22.33C24 [×2], C22.35C24 [×2], C22.36C24 [×2], C22.36C24 [×4], C23.41C23 [×2], C22.46C24 [×2], C22.46C24 [×2], C22.47C24 [×2], D43Q8, C22.49C24, C22.50C24, C22.50C24 [×2], Q83Q8, C22.56C24 [×2], C22.57C24 [×2], C22.154C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2- 1+4 [×2], C25, C2×2- 1+4, C2.C25 [×2], C22.154C25

Smallest permutation representation of C22.154C25
On 64 points
Generators in S64
(1 48)(2 45)(3 46)(4 47)(5 19)(6 20)(7 17)(8 18)(9 23)(10 24)(11 21)(12 22)(13 25)(14 26)(15 27)(16 28)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 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 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 30)(2 51)(3 32)(4 49)(5 35)(6 56)(7 33)(8 54)(9 59)(10 40)(11 57)(12 38)(13 61)(14 42)(15 63)(16 44)(17 53)(18 34)(19 55)(20 36)(21 37)(22 58)(23 39)(24 60)(25 41)(26 62)(27 43)(28 64)(29 47)(31 45)(46 52)(48 50)
(1 22 48 12)(2 21 45 11)(3 24 46 10)(4 23 47 9)(5 25 19 13)(6 28 20 16)(7 27 17 15)(8 26 18 14)(29 57 49 37)(30 60 50 40)(31 59 51 39)(32 58 52 38)(33 41 53 61)(34 44 54 64)(35 43 55 63)(36 42 56 62)
(1 20 46 8)(2 19 47 7)(3 18 48 6)(4 17 45 5)(9 27 21 13)(10 26 22 16)(11 25 23 15)(12 28 24 14)(29 53 51 35)(30 56 52 34)(31 55 49 33)(32 54 50 36)(37 41 59 63)(38 44 60 62)(39 43 57 61)(40 42 58 64)
(1 25 48 13)(2 14 45 26)(3 27 46 15)(4 16 47 28)(5 22 19 12)(6 9 20 23)(7 24 17 10)(8 11 18 21)(29 64 49 44)(30 41 50 61)(31 62 51 42)(32 43 52 63)(33 60 53 40)(34 37 54 57)(35 58 55 38)(36 39 56 59)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111111144
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C22- 1+4C2.C25
kernelC22.154C25C23.36C23C23.37C23C22.33C24C22.35C24C22.36C24C23.41C23C22.46C24C22.47C24D43Q8C22.49C24C22.50C24Q83Q8C22.56C24C22.57C24C4C2
# reps12122624211312224

Matrix representation of C22.154C25 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
00130000
00040000
42000000
01000000
00000031
00000002
00002400
00000300
,
00300000
00030000
20000000
02000000
00000040
00000004
00004000
00000400
,
00100000
00010000
10000000
01000000
00002000
00000200
00000020
00000002
,
13000000
14000000
00130000
00140000
00000010
00000001
00004000
00000400
,
40000000
04000000
00400000
00040000
00004300
00001100
00000043
00000011

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

C22.154C25 in GAP, Magma, Sage, TeX

C_2^2._{154}C_2^5
% in TeX

G:=Group("C2^2.154C2^5");
// GroupNames label

G:=SmallGroup(128,2297);
// by ID

G=gap.SmallGroup(128,2297);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,723,184,2019,570,360,1684,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=1,c^2=b,e^2=g^2=a,f^2=b*a=a*b,d*c*d=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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