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G = C23.241C24order 128 = 27

94th central extension by C23 of C24

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

Aliases: C23.241C24, C24.214C23, C22.552- 1+4, C22.752+ 1+4, C22.14(C4×D4), C2.5(D46D4), C2.7(D45D4), C22⋊C4.184D4, C23.418(C2×D4), C22.D49C4, (C2×C42).22C22, C23.8Q816C2, C23.293(C4○D4), C22.132(C23×C4), (C23×C4).308C22, (C22×C4).763C23, C23.132(C22×C4), C23.23D4.9C2, C22.112(C22×D4), C24.C2218C2, (C22×D4).486C22, C23.65C2324C2, C23.63C2317C2, C2.5(C22.45C24), C2.C42.62C22, C2.6(C22.46C24), C2.32(C23.33C23), (C4×C4⋊C4)⋊41C2, C2.35(C2×C4×D4), C4⋊C429(C2×C4), (C2×C4×D4).39C2, C2.33(C4×C4○D4), (C4×C22⋊C4)⋊40C2, C22⋊C415(C2×C4), (C22×C4)⋊34(C2×C4), (C2×C4).887(C2×D4), (C2×D4).169(C2×C4), (C2×C4).40(C22×C4), (C2×C42⋊C2)⋊14C2, (C2×C4).722(C4○D4), (C2×C4⋊C4).189C22, C22.126(C2×C4○D4), (C2×C2.C42)⋊21C2, (C2×C22.D4).7C2, (C2×C22⋊C4).555C22, C22⋊C42(C2.C42), SmallGroup(128,1091)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.241C24
C1C2C22C23C24C23×C4C2×C42⋊C2 — C23.241C24
C1C22 — C23.241C24
C1C23 — C23.241C24
C1C23 — C23.241C24

Generators and relations for C23.241C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, gag-1=ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >

Subgroups: 556 in 326 conjugacy classes, 152 normal (82 characteristic)
C1, C2 [×7], C2 [×6], C4 [×22], C22 [×7], C22 [×4], C22 [×22], C2×C4 [×18], C2×C4 [×50], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×9], C22⋊C4 [×16], C22⋊C4 [×7], C4⋊C4 [×8], C4⋊C4 [×9], C22×C4 [×13], C22×C4 [×4], C22×C4 [×19], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42 [×5], C2×C22⋊C4 [×8], C2×C4⋊C4 [×7], C42⋊C2 [×4], C4×D4 [×4], C22.D4 [×8], C23×C4 [×4], C22×D4, C2×C2.C42, C4×C22⋊C4 [×2], C4×C4⋊C4, C23.8Q8 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×2], C23.65C23, C2×C42⋊C2, C2×C4×D4, C2×C22.D4, C23.241C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×6], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C4×D4, C4×C4○D4, C23.33C23, D45D4, D46D4, C22.45C24, C22.46C24, C23.241C24

Smallest permutation representation of C23.241C24
On 64 points
Generators in S64
(5 64)(6 61)(7 62)(8 63)(9 34)(10 35)(11 36)(12 33)(21 27)(22 28)(23 25)(24 26)(29 54)(30 55)(31 56)(32 53)(37 46)(38 47)(39 48)(40 45)(41 59)(42 60)(43 57)(44 58)
(1 16)(2 13)(3 14)(4 15)(5 46)(6 47)(7 48)(8 45)(9 34)(10 35)(11 36)(12 33)(17 50)(18 51)(19 52)(20 49)(21 42)(22 43)(23 44)(24 41)(25 58)(26 59)(27 60)(28 57)(29 54)(30 55)(31 56)(32 53)(37 64)(38 61)(39 62)(40 63)
(1 20)(2 17)(3 18)(4 19)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 29)(12 30)(13 50)(14 51)(15 52)(16 49)(21 27)(22 28)(23 25)(24 26)(33 55)(34 56)(35 53)(36 54)(41 59)(42 60)(43 57)(44 58)(45 63)(46 64)(47 61)(48 62)
(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 43 16 22)(2 44 13 23)(3 41 14 24)(4 42 15 21)(5 9 46 34)(6 10 47 35)(7 11 48 36)(8 12 45 33)(17 58 50 25)(18 59 51 26)(19 60 52 27)(20 57 49 28)(29 62 54 39)(30 63 55 40)(31 64 56 37)(32 61 53 38)
(1 35 16 10)(2 11 13 36)(3 33 14 12)(4 9 15 34)(5 60 46 27)(6 28 47 57)(7 58 48 25)(8 26 45 59)(17 29 50 54)(18 55 51 30)(19 31 52 56)(20 53 49 32)(21 37 42 64)(22 61 43 38)(23 39 44 62)(24 63 41 40)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4T4U···4AJ
order12···22222224···44···4
size11···12222442···24···4

50 irreducible representations

dim111111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.241C24C2×C2.C42C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C2×C42⋊C2C2×C4×D4C2×C22.D4C22.D4C22⋊C4C2×C4C23C22C22
# reps1121212211111648411

Matrix representation of C23.241C24 in GL5(𝔽5)

40000
01000
01400
00010
00024
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
04000
00400
00010
00001
,
20000
03000
00300
00040
00031
,
40000
01300
00400
00020
00002
,
10000
01000
01400
00014
00024

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

C23.241C24 in GAP, Magma, Sage, TeX

C_2^3._{241}C_2^4
% in TeX

G:=Group("C2^3.241C2^4");
// GroupNames label

G:=SmallGroup(128,1091);
// by ID

G=gap.SmallGroup(128,1091);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346]);
// 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=b,g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,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*f=f*e>;
// generators/relations

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