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G = C2×C22.49C24  order 128 = 27

Direct product of C2 and C22.49C24

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

Aliases: C2×C22.49C24, C23.28C24, C22.62C25, C24.496C23, C42.560C23, C22.1142+ (1+4), C4⋊Q8⋊85C22, (C2×C4).61C24, (C4×D4)⋊108C22, C4⋊C4.518C23, C4⋊D4⋊75C22, (C2×D4).457C23, C4.4D4⋊74C22, C22⋊C4.86C23, (C2×Q8).280C23, C42⋊C2⋊97C22, (C2×C42).931C22, (C23×C4).600C22, C2.21(C2×2+ (1+4)), (C22×C4).1198C23, (C22×D4).423C22, (C22×Q8).355C22, (C2×C4×D4)⋊86C2, (C2×C4⋊Q8)⋊53C2, (C2×C4⋊D4)⋊64C2, C4.135(C2×C4○D4), (C2×C4.4D4)⋊52C2, C2.34(C22×C4○D4), (C2×C42⋊C2)⋊62C2, (C2×C4).851(C4○D4), (C2×C4⋊C4).983C22, C22.160(C2×C4○D4), (C2×C22⋊C4).542C22, SmallGroup(128,2205)

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

Derived series
C1 — C22 — C2×C22.49C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.49C24
Lower central
C1 — C22 — C2×C22.49C24
Upper central
C1 — C23 — C2×C22.49C24
Jennings
C1 — C22 — C2×C22.49C24

Subgroups: 956 in 620 conjugacy classes, 404 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×8], C4 [×18], C22, C22 [×6], C22 [×40], C2×C4 [×30], C2×C4 [×50], D4 [×32], Q8 [×8], C23, C23 [×8], C23 [×24], C42 [×20], C22⋊C4 [×48], C4⋊C4 [×24], C22×C4, C22×C4 [×26], C22×C4 [×16], C2×D4 [×24], C2×D4 [×16], C2×Q8 [×8], C2×Q8 [×4], C24 [×4], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C42⋊C2 [×32], C4×D4 [×16], C4⋊D4 [×32], C4.4D4 [×32], C4⋊Q8 [×8], C23×C4 [×4], C22×D4 [×6], C22×Q8 [×2], C2×C42⋊C2 [×4], C2×C4×D4 [×2], C2×C4⋊D4 [×4], C2×C4.4D4 [×4], C2×C4⋊Q8, C22.49C24 [×16], C2×C22.49C24

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], 2+ (1+4) [×2], C25, C22.49C24 [×4], C22×C4○D4 [×2], C2×2+ (1+4), C2×C22.49C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

40000
04000
00400
00040
00004
,
10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
10000
01000
00400
00013
00004
,
40000
00100
01000
00030
00003
,
40000
02000
00200
00040
00041
,
40000
03000
00200
00040
00004

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

50 conjugacy classes

class 1 2A···2G2H···2O4A···4X4Y···4AH
order12···22···24···44···4
size11···14···42···24···4

50 irreducible representations

dim111111124
type++++++++
imageC1C2C2C2C2C2C2C4○D42+ (1+4)
kernelC2×C22.49C24C2×C42⋊C2C2×C4×D4C2×C4⋊D4C2×C4.4D4C2×C4⋊Q8C22.49C24C2×C4C22
# reps14244116162

In GAP, Magma, Sage, TeX 

C_2\times C_2^2._{49}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.49C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2205);
 
// by ID
 
G=gap.SmallGroup(128,2205);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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