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G = C23.225C24order 128 = 27

78th central extension by C23 of C24

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

Aliases: C23.225C24, C24.552C23, C22.622+ 1+4, C22.442- 1+4, C4222(C2×C4), C425C45C2, C42⋊C226C4, C424C413C2, C428C415C2, (C2×C42).16C22, C23.8Q8.6C2, C23.288(C4○D4), (C23×C4).299C22, (C22×C4).754C23, C22.116(C23×C4), C23.128(C22×C4), C23.34D4.10C2, C23.63C2312C2, C2.2(C22.45C24), C22.12(C42⋊C2), C2.C42.57C22, C2.1(C22.46C24), C2.24(C23.33C23), (C4×C4⋊C4)⋊31C2, C4⋊C442(C2×C4), C2.25(C4×C4○D4), (C4×C22⋊C4).6C2, C22⋊C4.84(C2×C4), (C2×C4).719(C4○D4), (C2×C4⋊C4).818C22, (C2×C4).230(C22×C4), (C22×C4).308(C2×C4), C2.26(C2×C42⋊C2), C22.110(C2×C4○D4), (C2×C42⋊C2).30C2, (C2×C22⋊C4).554C22, (C2×C2.C42).20C2, SmallGroup(128,1075)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.225C24
C1C2C22C23C24C23×C4C2×C42⋊C2 — C23.225C24
C1C22 — C23.225C24
C1C23 — C23.225C24
C1C23 — C23.225C24

Generators and relations for C23.225C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=b, f2=a, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 428 in 266 conjugacy classes, 144 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×22], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×16], C2×C4 [×50], C23, C23 [×6], C23 [×4], C42 [×8], C42 [×6], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×6], C22×C4 [×12], C22×C4 [×14], C24, C2.C42 [×4], C2.C42 [×12], C2×C42 [×2], C2×C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×8], C23×C4, C23×C4 [×2], C2×C2.C42, C424C4, C4×C22⋊C4 [×2], C4×C4⋊C4, C23.34D4, C428C4, C425C4, C23.8Q8 [×2], C23.63C23 [×4], C2×C42⋊C2, C23.225C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C42⋊C2, C4×C4○D4, C23.33C23, C22.45C24 [×2], C22.46C24 [×2], C23.225C24

Smallest permutation representation of C23.225C24
On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 51 47)(2 32 52 20)(3 57 49 45)(4 30 50 18)(5 42 36 54)(6 15 33 27)(7 44 34 56)(8 13 35 25)(9 29 21 17)(10 58 22 46)(11 31 23 19)(12 60 24 48)(14 62 26 38)(16 64 28 40)(37 41 61 53)(39 43 63 55)
(1 15 11 43)(2 28 12 56)(3 13 9 41)(4 26 10 54)(5 30 38 58)(6 19 39 47)(7 32 40 60)(8 17 37 45)(14 22 42 50)(16 24 44 52)(18 62 46 36)(20 64 48 34)(21 53 49 25)(23 55 51 27)(29 61 57 35)(31 63 59 33)
(1 3)(2 4)(5 40)(6 37)(7 38)(8 39)(9 11)(10 12)(13 15)(14 16)(17 47)(18 48)(19 45)(20 46)(21 23)(22 24)(25 27)(26 28)(29 59)(30 60)(31 57)(32 58)(33 61)(34 62)(35 63)(36 64)(41 43)(42 44)(49 51)(50 52)(53 55)(54 56)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim1111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4C4○D4C4○D42+ 1+42- 1+4
kernelC23.225C24C2×C2.C42C424C4C4×C22⋊C4C4×C4⋊C4C23.34D4C428C4C425C4C23.8Q8C23.63C23C2×C42⋊C2C42⋊C2C2×C4C23C22C22
# reps11121111241168811

Matrix representation of C23.225C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
040000
002000
003300
000010
000044
,
200000
020000
003000
000300
000012
000004
,
010000
100000
004300
000100
000030
000003
,
100000
010000
004000
000400
000010
000044

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

C23.225C24 in GAP, Magma, Sage, TeX

C_2^3._{225}C_2^4
% in TeX

G:=Group("C2^3.225C2^4");
// GroupNames label

G:=SmallGroup(128,1075);
// by ID

G=gap.SmallGroup(128,1075);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,346]);
// Polycyclic

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

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