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G = C22.50C25order 128 = 27

31st central stem extension by C22 of C25

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

Aliases: C22.50C25, C23.122C24, C42.551C23, C4.1312- 1+4, (C2×C4).52C24, C4⋊C4.464C23, C4⋊Q8.332C22, (C2×D4).296C23, (C4×D4).349C22, C22⋊C4.80C23, (C4×Q8).319C22, (C2×Q8).428C23, C4⋊D4.219C22, C2.9(C2.C25), (C2×C42).923C22, C22⋊Q8.224C22, C2.10(C2×2- 1+4), (C22×C4).1189C23, C422C2.13C22, C4.4D4.169C22, C42.C2.146C22, (C22×Q8).490C22, C43(C23.41C23), C43(C22.36C24), C43(C22.33C24), C42(C23.38C23), C43(C22.31C24), C42(C22.35C24), C42⋊C2.221C22, C23.36C2320C2, C23.41C2330C2, C22.33C2430C2, C22.36C2447C2, C22.35C2429C2, C23.38C2340C2, C23.37C2330C2, C22.D4.25C22, C22.31C24.11C2, (C2×C4×Q8)⋊48C2, (C4×C4○D4)⋊19C2, C4.77(C2×C4○D4), C22.13(C2×C4○D4), C2.24(C22×C4○D4), (C2×C4).305(C4○D4), (C2×C4⋊C4).953C22, (C2×C4○D4).323C22, (C2×C4)(C23.38C23), SmallGroup(128,2193)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.50C25
C1C2C22C2×C4C22×C4C2×C42C2×C4×Q8 — C22.50C25
C1C22 — C22.50C25
C1C2×C4 — C22.50C25
C1C22 — C22.50C25

Generators and relations for C22.50C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=a, g2=b, ab=ba, dcd-1=fcf=ac=ca, ede-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 660 in 506 conjugacy classes, 390 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×23], C22, C22 [×2], C22 [×14], C2×C4 [×2], C2×C4 [×28], C2×C4 [×29], D4 [×16], Q8 [×16], C23, C23 [×4], C42 [×24], C22⋊C4 [×36], C4⋊C4 [×56], C22×C4 [×3], C22×C4 [×16], C2×D4 [×10], C2×Q8 [×10], C2×Q8 [×4], C4○D4 [×8], C2×C42, C2×C42 [×6], C2×C4⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×14], C4×D4 [×16], C4×Q8 [×16], C4⋊D4 [×8], C22⋊Q8 [×24], C22.D4 [×16], C4.4D4 [×8], C42.C2 [×16], C422C2 [×16], C4⋊Q8 [×8], C22×Q8, C2×C4○D4 [×2], C2×C4×Q8, C4×C4○D4 [×2], C23.36C23 [×8], C23.37C23 [×4], C23.38C23 [×2], C22.31C24, C22.33C24 [×4], C22.35C24 [×4], C22.36C24 [×4], C23.41C23, C22.50C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2- 1+4 [×2], C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.50C25

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O···4AH
order122222222244444···44···4
size111122444411112···24···4

44 irreducible representations

dim11111111111244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.50C25C2×C4×Q8C4×C4○D4C23.36C23C23.37C23C23.38C23C22.31C24C22.33C24C22.35C24C22.36C24C23.41C23C2×C4C4C2
# reps11284214441822

Matrix representation of C22.50C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
030000
200000
000300
002000
003324
002033
,
100000
010000
003324
000030
000300
000002
,
010000
100000
000010
001143
004000
001104
,
100000
010000
000100
001000
001143
000001
,
300000
030000
001000
000100
000010
000001

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,3,2,0,0,3,0,3,0,0,0,0,0,2,3,0,0,0,0,4,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,3,0,3,0,0,0,2,3,0,0,0,0,4,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,4,1,0,0,0,1,0,1,0,0,1,4,0,0,0,0,0,3,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,4,0,0,0,0,0,3,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22.50C25 in GAP, Magma, Sage, TeX

C_2^2._{50}C_2^5
% in TeX

G:=Group("C2^2.50C2^5");
// GroupNames label

G:=SmallGroup(128,2193);
// by ID

G=gap.SmallGroup(128,2193);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,387,184,1123,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=a,g^2=b,a*b=b*a,d*c*d^-1=f*c*f=a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

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