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G = C24.205C23  order 128 = 27

45th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.205C23, C23.219C24, C22.572+ 1+4, C22.402- 1+4, C42⋊21(C2×C4), C42⋊5C4⋊4C2, C4.4D4⋊23C4, (C23×C4).51C22, C23.17(C22×C4), C23.8Q8⋊11C2, (C22×C4).484C23, C22.110(C23×C4), (C2×C42).423C22, C23.23D4.6C2, (C22×Q8).90C22, C24.C22⋊11C2, C2.9(C22.32C24), (C22×D4).109C22, C23.67C23⋊17C2, C2.22(C22.11C24), C2.C42.54C22, C2.21(C23.33C23), C2.11(C22.36C24), (C4×C4⋊C4)⋊30C2, (C2×Q8)⋊14(C2×C4), C2.22(C4×C4○D4), C22⋊C4⋊14(C2×C4), (C4×C22⋊C4)⋊35C2, (C2×D4).130(C2×C4), (C2×C4).521(C4○D4), (C2×C4⋊C4).814C22, (C2×C4).227(C22×C4), (C2×C4.4D4).16C2, C22.104(C2×C4○D4), (C2×C22⋊C4).432C22, SmallGroup(128,1069)

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

Derived series
C1 — C22 — C24.205C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.205C23
Lower central
C1 — C22 — C24.205C23
Upper central
C1 — C23 — C24.205C23
Jennings
C1 — C23 — C24.205C23

Generators and relations for C24.205C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=db=bd, f2=b, g2=c, gag-1=ab=ba, ac=ca, ad=da, ae=ea, faf-1=abc, bc=cb, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, gfg-1=bcf >

Subgroups: 508 in 270 conjugacy classes, 136 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C42⋊5C4, C23.8Q8, C23.23D4, C24.C22, C23.67C23, C2×C4.4D4, C24.205C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, 2+ 1+4, 2- 1+4, C4×C4○D4, C22.11C24, C23.33C23, C22.32C24, C22.36C24, C24.205C23

Smallest permutation representation of C24.205C23
►On 64 points
Generators in S64
(5 13)(6 14)(7 15)(8 16)(9 46)(10 47)(11 48)(12 45)(17 33)(18 34)(19 35)(20 36)(21 63)(22 64)(23 61)(24 62)(25 57)(26 58)(27 59)(28 60)(37 52)(38 49)(39 50)(40 51)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4AF
order12···222224···44···4
size11···144442···24···4

44 irreducible representations

dim1111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC24.205C23C4×C22⋊C4C4×C4⋊C4C42⋊5C4C23.8Q8C23.23D4C24.C22C23.67C23C2×C4.4D4C4.4D4C2×C4C22C22
# reps12112242116831

Matrix representation of C24.205C23 ►in GL8(𝔽5)

10000000
44000000
00100000
00340000
00001000
00002400
00000040
00000031
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00200000
00020000
00000023
00000003
00002300
00000300
,
43000000
01000000
00220000
00130000
00000040
00000004
00001000
00000100
,
20000000
33000000
00300000
00420000
00003200
00001200
00000023
00000043

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

C24.205C23 in GAP, Magma, Sage, TeX 

C_2^4._{205}C_2^3
 
% in TeX
 
G:=Group("C2^4.205C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1069);
 
// by ID
 
G=gap.SmallGroup(128,1069);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,219,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*b=b*d,f^2=b,g^2=c,g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*c,b*c=c*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,g*f*g^-1=b*c*f>;
 
// generators/relations
 

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