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G = C24.204C23  order 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, C22.552+ (1+4), C22.382- (1+4), C22.D4⋊8C4, C23.16(C22×C4), (C2×C42).15C22, C23.7Q8⋊18C2, C23.8Q8⋊10C2, (C23×C4).295C22, C22.108(C23×C4), (C22×C4).482C23, C23.23D4.5C2, C24.C22⋊10C2, C2.8(C22.32C24), (C22×D4).108C22, C23.65C23⋊17C2, C23.63C23⋊11C2, C2.21(C22.11C24), C2.C42.52C22, C24.3C22.26C2, C2.7(C22.34C24), C2.8(C22.33C24), C2.10(C22.36C24), C2.19(C23.33C23), (C4×C4⋊C4)⋊29C2, C4⋊C4⋊14(C2×C4), C2.20(C4×C4○D4), (C4×C22⋊C4)⋊34C2, C22⋊C4⋊13(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

Derived series
C1 — C22 — C24.204C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.204C23
Lower central
C1 — C22 — C24.204C23
Upper central
C1 — C23 — C24.204C23
Jennings
C1 — C23 — C24.204C23

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

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC24.204C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C2×C22.D4C22.D4C2×C4C22C22
# reps1211112411116831

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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