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G = C24.204C23order 128 = 27

44th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.204C23, C23.217C24, C22.382- 1+4, C22.552+ 1+4, C22.D48C4, C23.16(C22×C4), (C2×C42).15C22, C23.7Q818C2, C23.8Q810C2, C22.108(C23×C4), (C22×C4).482C23, (C23×C4).295C22, C23.23D4.5C2, C24.C2210C2, C2.8(C22.32C24), (C22×D4).108C22, C23.63C2311C2, C23.65C2317C2, C2.21(C22.11C24), C24.3C22.26C2, C2.C42.52C22, C2.8(C22.33C24), C2.7(C22.34C24), C2.19(C23.33C23), C2.10(C22.36C24), (C4×C4⋊C4)⋊29C2, C4⋊C414(C2×C4), C2.20(C4×C4○D4), (C4×C22⋊C4)⋊34C2, C22⋊C413(C2×C4), (C22×C4)⋊27(C2×C4), (C2×D4).129(C2×C4), (C2×C4).37(C22×C4), (C2×C4).519(C4○D4), (C2×C4⋊C4).812C22, C22.102(C2×C4○D4), (C2×C22⋊C4).31C22, (C2×C22.D4).6C2, SmallGroup(128,1067)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.204C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.204C23
C1C22 — C24.204C23
C1C23 — C24.204C23
C1C23 — C24.204C23

Generators and relations for C24.204C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, g2=cb=bc, faf-1=ab=ba, eae-1=ac=ca, ad=da, ag=ga, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 492 in 266 conjugacy classes, 136 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×20], C22 [×7], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×6], C22⋊C4 [×12], C22⋊C4 [×10], C4⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×13], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42 [×5], C2×C22⋊C4 [×10], C2×C4⋊C4 [×7], C22.D4 [×8], C23×C4 [×2], C22×D4, C4×C22⋊C4 [×2], C4×C4⋊C4, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23 [×2], C24.C22 [×4], C23.65C23, C24.3C22, C2×C22.D4, C24.204C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C4×C4○D4, C22.11C24, C23.33C23, C22.32C24, C22.33C24, C22.34C24, C22.36C24, C24.204C23

Smallest permutation representation of C24.204C23
On 64 points
Generators in S64
(1 14)(2 39)(3 16)(4 37)(5 46)(6 19)(7 48)(8 17)(9 56)(10 34)(11 54)(12 36)(13 64)(15 62)(18 26)(20 28)(21 60)(22 29)(23 58)(24 31)(25 45)(27 47)(30 49)(32 51)(33 41)(35 43)(38 61)(40 63)(42 53)(44 55)(50 59)(52 57)
(1 30)(2 31)(3 32)(4 29)(5 54)(6 55)(7 56)(8 53)(9 48)(10 45)(11 46)(12 47)(13 52)(14 49)(15 50)(16 51)(17 42)(18 43)(19 44)(20 41)(21 40)(22 37)(23 38)(24 39)(25 34)(26 35)(27 36)(28 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 26)(6 27)(7 28)(8 25)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 45)(18 46)(19 47)(20 48)(21 51)(22 52)(23 49)(24 50)(29 57)(30 58)(31 59)(32 60)(33 56)(34 53)(35 54)(36 55)
(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 20 3 18)(2 42 4 44)(5 23 7 21)(6 39 8 37)(9 60 11 58)(10 64 12 62)(13 27 15 25)(14 33 16 35)(17 29 19 31)(22 55 24 53)(26 49 28 51)(30 41 32 43)(34 52 36 50)(38 56 40 54)(45 57 47 59)(46 61 48 63)
(1 25 58 53)(2 35 59 5)(3 27 60 55)(4 33 57 7)(6 32 36 63)(8 30 34 61)(9 22 20 13)(10 38 17 49)(11 24 18 15)(12 40 19 51)(14 45 23 42)(16 47 21 44)(26 62 54 31)(28 64 56 29)(37 41 52 48)(39 43 50 46)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim111111111111244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC24.204C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C2×C22.D4C22.D4C2×C4C22C22
# reps1211112411116831

Matrix representation of C24.204C23 in GL8(𝔽5)

31000000
22000000
00130000
00040000
00000010
00000001
00001000
00000100
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
11000000
00420000
00410000
00000001
00000040
00000400
00001000
,
10000000
01000000
00200000
00020000
00004000
00000400
00000010
00000001
,
20000000
02000000
00300000
00030000
00000100
00001000
00000001
00000010

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

C24.204C23 in GAP, Magma, Sage, TeX

C_2^4._{204}C_2^3
% in TeX

G:=Group("C2^4.204C2^3");
// GroupNames label

G:=SmallGroup(128,1067);
// by ID

G=gap.SmallGroup(128,1067);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,675,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d,g^2=c*b=b*c,f*a*f^-1=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*d=d*b,f*e*f^-1=g*e*g^-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,f*g=g*f>;
// generators/relations

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