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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), (C4×Q8)⋊81C22, (C22×Q8)⋊24C4, C4⋊C4.513C23, (C2×C4).158C24, Q8.24(C22×C4), (Q8×C23).13C2, 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, (C2×C4).282(C22×C4), (C22×C4).372(C2×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

Generators and relations for C2×C23.32C23
 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 >

Subgroups: 796 in 744 conjugacy classes, 692 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C23×C4, C22×Q8, C2×C42⋊C2, C2×C4×Q8, C23.32C23, Q8×C23, C2×C23.32C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2- 1+4, C25, C23.32C23, C24×C4, C2×2- 1+4, C2×C23.32C23

Smallest permutation representation of C2×C23.32C23
►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 55 19 9)(2 56 20 10)(3 53 17 11)(4 54 18 12)(5 25 61 47)(6 26 62 48)(7 27 63 45)(8 28 64 46)(13 31 51 41)(14 32 52 42)(15 29 49 43)(16 30 50 44)(21 35 59 37)(22 36 60 38)(23 33 57 39)(24 34 58 40)
 
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), (1,55,19,9)(2,56,20,10)(3,53,17,11)(4,54,18,12)(5,25,61,47)(6,26,62,48)(7,27,63,45)(8,28,64,46)(13,31,51,41)(14,32,52,42)(15,29,49,43)(16,30,50,44)(21,35,59,37)(22,36,60,38)(23,33,57,39)(24,34,58,40)>;
 
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), (1,55,19,9)(2,56,20,10)(3,53,17,11)(4,54,18,12)(5,25,61,47)(6,26,62,48)(7,27,63,45)(8,28,64,46)(13,31,51,41)(14,32,52,42)(15,29,49,43)(16,30,50,44)(21,35,59,37)(22,36,60,38)(23,33,57,39)(24,34,58,40) );
 
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)], [(1,55,19,9),(2,56,20,10),(3,53,17,11),(4,54,18,12),(5,25,61,47),(6,26,62,48),(7,27,63,45),(8,28,64,46),(13,31,51,41),(14,32,52,42),(15,29,49,43),(16,30,50,44),(21,35,59,37),(22,36,60,38),(23,33,57,39),(24,34,58,40)]])
 

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

Matrix representation of C2×C23.32C23 ►in 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] >;
 

C2×C23.32C23 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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