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G = C22.98C25order 128 = 27

79th central stem extension by C22 of C25

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

Aliases: C22.98C25, C42.90C23, C23.48C24, C4.1072- 1+4, Q83Q819C2, (C2×C4).88C24, C4⋊C4.493C23, C4⋊Q8.358C22, (C2×D4).474C23, (C4×D4).358C22, C22⋊C4.25C23, (C4×Q8).223C22, (C2×Q8).453C23, (C2×C42).950C22, C422C2.2C22, C4.4D4.96C22, C22⋊Q8.229C22, C2.27(C2×2- 1+4), C42.C2.82C22, (C22×C4).1210C23, (C22×Q8).361C22, C22.D4.9C22, C42⋊C2.230C22, C23.37C2341C2, C22.46C2418C2, C22.35C2410C2, C22.50C2422C2, C23.32C2314C2, C23.38C23.16C2, (C2×C4⋊Q8)⋊57C2, C22⋊C4(C4⋊Q8), (C4×C4○D4).31C2, C4.181(C2×C4○D4), C22.31(C2×C4○D4), C2.54(C22×C4○D4), (C2×C4).307(C4○D4), (C2×C4⋊C4).707C22, (C2×C4○D4).330C22, SmallGroup(128,2241)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.98C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.98C25
C1C22 — C22.98C25
C1C22 — C22.98C25
C1C22 — C22.98C25

Generators and relations for C22.98C25
 G = < a,b,c,d,e,f,g | a2=b2=f2=1, c2=g2=a, d2=e2=b, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 604 in 496 conjugacy classes, 392 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C4 [×24], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×24], D4 [×6], Q8 [×26], C23, C23 [×2], C42 [×2], C42 [×28], C22⋊C4 [×26], C4⋊C4 [×70], C22×C4, C22×C4 [×12], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×12], C2×Q8 [×8], C4○D4 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×4], C42⋊C2, C42⋊C2 [×20], C4×D4 [×6], C4×Q8 [×30], C22⋊Q8 [×20], C22.D4 [×8], C4.4D4 [×4], C42.C2 [×26], C422C2 [×16], C4⋊Q8 [×14], C22×Q8 [×2], C2×C4○D4, C4×C4○D4, C23.32C23 [×2], C2×C4⋊Q8, C23.37C23, C23.37C23 [×4], C23.38C23 [×2], C22.35C24 [×4], C22.46C24 [×8], C22.50C24 [×4], Q83Q8 [×4], C22.98C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2- 1+4 [×4], C25, C22×C4○D4, C2×2- 1+4 [×2], C22.98C25

Smallest permutation representation of C22.98C25
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 15 51 59)(2 14 52 58)(3 13 49 57)(4 16 50 60)(5 27 36 43)(6 26 33 42)(7 25 34 41)(8 28 35 44)(9 19 53 63)(10 18 54 62)(11 17 55 61)(12 20 56 64)(21 29 37 45)(22 32 38 48)(23 31 39 47)(24 30 40 46)
(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 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 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), (1,15,51,59)(2,14,52,58)(3,13,49,57)(4,16,50,60)(5,27,36,43)(6,26,33,42)(7,25,34,41)(8,28,35,44)(9,19,53,63)(10,18,54,62)(11,17,55,61)(12,20,56,64)(21,29,37,45)(22,32,38,48)(23,31,39,47)(24,30,40,46), (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), (1,11)(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), (1,37,3,39)(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), (1,15,51,59)(2,14,52,58)(3,13,49,57)(4,16,50,60)(5,27,36,43)(6,26,33,42)(7,25,34,41)(8,28,35,44)(9,19,53,63)(10,18,54,62)(11,17,55,61)(12,20,56,64)(21,29,37,45)(22,32,38,48)(23,31,39,47)(24,30,40,46), (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), (1,11)(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), (1,37,3,39)(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)], [(1,15,51,59),(2,14,52,58),(3,13,49,57),(4,16,50,60),(5,27,36,43),(6,26,33,42),(7,25,34,41),(8,28,35,44),(9,19,53,63),(10,18,54,62),(11,17,55,61),(12,20,56,64),(21,29,37,45),(22,32,38,48),(23,31,39,47),(24,30,40,46)], [(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)], [(1,11),(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)], [(1,37,3,39),(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)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4P4Q···4AJ
order122222224···44···4
size111122442···24···4

44 irreducible representations

dim111111111124
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4○D42- 1+4
kernelC22.98C25C4×C4○D4C23.32C23C2×C4⋊Q8C23.37C23C23.38C23C22.35C24C22.46C24C22.50C24Q83Q8C2×C4C4
# reps112152484484

Matrix representation of C22.98C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
100000
003020
003423
000020
003141
,
300000
030000
004000
004100
003010
002414
,
200000
030000
004000
000400
000040
000004
,
400000
040000
004200
000100
003113
000404
,
100000
010000
002000
000200
004030
000303

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

C22.98C25 in GAP, Magma, Sage, TeX

C_2^2._{98}C_2^5
% in TeX

G:=Group("C2^2.98C2^5");
// GroupNames label

G:=SmallGroup(128,2241);
// by ID

G=gap.SmallGroup(128,2241);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,352,570,1684,102]);
// Polycyclic

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

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