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G = C2×C23.41C23  order 128 = 27

Direct product of C2 and C23.41C23

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

Aliases: C2×C23.41C23, C22.46C25, C24.614C23, C23.118C24, C42.547C23, C22.782- (1+4), C22.1072+ (1+4), C4⋊Q8⋊78C22, (C22×C4)⋊17Q8, C2.8(Q8×C23), (C2×C4).48C24, C4.19(C22×Q8), C4⋊C4.286C23, C23.110(C2×Q8), C22.6(C22×Q8), C22⋊C4.76C23, (C2×Q8).276C23, C42.C2⋊43C22, C2.9(C2×2- (1+4)), (C2×C42).921C22, (C23×C4).590C22, C22⋊Q8.222C22, C2.12(C2×2+ (1+4)), (C22×C4).1586C23, (C22×Q8).353C22, C42⋊C2.339C22, (C2×C4)⋊6(C2×Q8), (C2×C4⋊Q8)⋊49C2, (C22×C4⋊C4).49C2, (C2×C42.C2)⋊41C2, (C2×C22⋊Q8).61C2, (C2×C4⋊C4).701C22, (C2×C42⋊C2).64C2, (C2×C22⋊C4).535C22, SmallGroup(128,2189)

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

Derived series
C1 — C22 — C2×C23.41C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C2×C23.41C23
Lower central
C1 — C22 — C2×C23.41C23
Upper central
C1 — C23 — C2×C23.41C23
Jennings
C1 — C22 — C2×C23.41C23

Subgroups: 716 in 540 conjugacy classes, 436 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×24], C22, C22 [×10], C22 [×12], C2×C4 [×52], C2×C4 [×32], Q8 [×16], C23, C23 [×6], C23 [×4], C42 [×16], C22⋊C4 [×16], C4⋊C4 [×80], C22×C4 [×34], C22×C4 [×4], C2×Q8 [×16], C2×Q8 [×8], C24, C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×28], C42⋊C2 [×16], C22⋊Q8 [×32], C42.C2 [×32], C4⋊Q8 [×32], C23×C4, C23×C4 [×2], C22×Q8 [×4], C22×C4⋊C4, C2×C42⋊C2 [×2], C2×C22⋊Q8 [×4], C2×C42.C2 [×4], C2×C4⋊Q8 [×4], C23.41C23 [×16], C2×C23.41C23

Quotients:
C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], 2+ (1+4) [×2], 2- (1+4) [×2], C25, C23.41C23 [×4], Q8×C23, C2×2+ (1+4), C2×2- (1+4), C2×C23.41C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd >

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

(2 20)(4 18)(6 62)(8 64)(10 56)(12 54)(14 52)(16 50)(22 60)(24 58)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00003040
00000204
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
03000000
30000000
00030000
00300000
00002020
00000302
00000030
00000002
,
40000000
04000000
00100000
00010000
00000200
00002000
00000003
00000030
,
01000000
40000000
00010000
00400000
00000400
00001000
00000001
00000040

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,3,0,0,0,0,0,0,2,0,2],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF
order12···222224···44···4
size11···122222···24···4

44 irreducible representations

dim1111111244
type+++++++-+-
imageC1C2C2C2C2C2C2Q82+ (1+4)2- (1+4)
kernelC2×C23.41C23C22×C4⋊C4C2×C42⋊C2C2×C22⋊Q8C2×C42.C2C2×C4⋊Q8C23.41C23C22×C4C22C22
# reps11244416822

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{41}C_2^3
 
% in TeX
 
G:=Group("C2xC2^3.41C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2189);
 
// by ID
 
G=gap.SmallGroup(128,2189);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,1430,387,352,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,g*e*g^-1=d*e=e*d,d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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