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G = C2×C23.32C23  order 128 = 27

Direct product of C2 and C23.32C23

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

Aliases: C2×C23.32C23, C22.12C25, C42.534C23, C23.106C24, C24.604C23, C22.712- (1+4), C2.8(C24×C4), C4.38(C23×C4), (C22×Q8)⋊24C4, (C4×Q8)⋊81C22, C4⋊C4.513C23, (C2×C4).158C24, (Q8×C23).13C2, Q8.24(C22×C4), C22.19(C23×C4), (C2×Q8).479C23, C2.1(C2×2- (1+4)), C22⋊C4.125C23, C23.234(C22×C4), (C23×C4).576C22, (C2×C42).912C22, (C22×C4).1293C23, (C22×Q8).481C22, C42⋊C2.335C22, (C2×C4×Q8)⋊41C2, Q8○(C2×C22⋊C4), C22⋊C4○2(C2×Q8), (C2×Q8)⋊43(C2×C4), (C2×C4⋊C4).982C22, (C22×C4).372(C2×C4), (C2×C4).282(C22×C4), (C2×C42⋊C2).61C2, (C2×C22⋊C4).561C22, SmallGroup(128,2158)

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

Derived series
C1 — C2 — C2×C23.32C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — Q8×C23 — C2×C23.32C23
Lower central
C1 — C2 — C2×C23.32C23
Upper central
C1 — C23 — C2×C23.32C23
Jennings
C1 — C22 — C2×C23.32C23

Subgroups: 796 in 744 conjugacy classes, 692 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×24], C4 [×16], C22, C22 [×10], C22 [×12], C2×C4 [×100], C2×C4 [×16], Q8 [×64], C23, C23 [×6], C23 [×4], C42 [×48], C22⋊C4 [×16], C4⋊C4 [×48], C22×C4 [×50], C2×Q8 [×112], C24, C2×C42 [×12], C2×C22⋊C4 [×4], C2×C4⋊C4 [×12], C42⋊C2 [×48], C4×Q8 [×64], C23×C4 [×3], C22×Q8 [×28], C2×C42⋊C2 [×6], C2×C4×Q8 [×8], C23.32C23 [×16], Q8×C23, C2×C23.32C23

Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2- (1+4) [×4], C25, C23.32C23 [×4], C24×C4, C2×2- (1+4) [×2], C2×C23.32C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=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, de=ed, df=fd, dg=gd, eg=ge >

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

(2 20)(4 18)(5 40)(7 38)(10 54)(12 56)(14 50)(16 52)(22 58)(24 60)(26 46)(28 48)(30 42)(32 44)(34 62)(36 64)

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
020000
000010
000001
001000
000100
,
400000
040000
000200
002000
000003
000030
,
100000
010000
000100
004000
000001
000040

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4BD
order12···222224···4
size11···122222···2

68 irreducible representations

dim1111114
type+++++-
imageC1C2C2C2C2C42- (1+4)
kernelC2×C23.32C23C2×C42⋊C2C2×C4×Q8C23.32C23Q8×C23C22×Q8C22
# reps168161324

In GAP, Magma, Sage, TeX 

C_2\times C_2^3._{32}C_2^3
 
% in TeX
 
G:=Group("C2xC2^3.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2158);
 
// by ID
 
G=gap.SmallGroup(128,2158);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,232,387,184,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^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,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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