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G = C24.377C23  order 128 = 27

217th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.377C23, C23.561C24, C22.3352+ 1+4, (C22×C4)⋊39D4, C23⋊2D4⋊30C2, C23.200(C2×D4), C23.10D4⋊66C2, C23.11D4⋊71C2, C2.33(C23⋊3D4), (C22×C4).166C23, (C23×C4).437C22, (C2×C42).625C22, C22.373(C22×D4), C24.3C22⋊69C2, (C22×D4).209C22, C23.81C23⋊72C2, C24.C22⋊111C2, C2.49(C22.29C24), C2.52(C22.32C24), C2.C42.275C22, C2.50(C22.26C24), C2.36(C22.34C24), (C2×C4⋊D4)⋊26C2, (C4×C22⋊C4)⋊98C2, (C2×C4).683(C2×D4), (C2×C4).181(C4○D4), (C2×C4⋊C4).384C22, C22.428(C2×C4○D4), (C2×C22.D4)⋊27C2, (C2×C22⋊C4).520C22, SmallGroup(128,1393)

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

Derived series
C1 — C23 — C24.377C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.3C22 — C24.377C23
Lower central
C1 — C23 — C24.377C23
Upper central
C1 — C23 — C24.377C23
Jennings
C1 — C23 — C24.377C23

Generators and relations for C24.377C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=c, eae-1=ab=ba, faf=ac=ca, ad=da, ag=ga, bc=cb, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 692 in 299 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C22.D4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C24.C22, C24.3C22, C23⋊2D4, C23⋊2D4, C23.10D4, C23.10D4, C23.11D4, C23.81C23, C2×C4⋊D4, C2×C22.D4, C24.377C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.26C24, C23⋊3D4, C22.29C24, C22.32C24, C22.34C24, C24.377C23

Smallest permutation representation of C24.377C23
►On 64 points
Generators in S64
(1 55)(2 52)(3 53)(4 50)(5 36)(6 44)(7 34)(8 42)(9 45)(10 59)(11 47)(12 57)(13 54)(14 51)(15 56)(16 49)(17 39)(18 29)(19 37)(20 31)(21 33)(22 41)(23 35)(24 43)(25 48)(26 58)(27 46)(28 60)(30 62)(32 64)(38 63)(40 61)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4N4O···4S
order12···22222244444···44···4
size11···14488822224···48···8

32 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC24.377C23C4×C22⋊C4C24.C22C24.3C22C23⋊2D4C23.10D4C23.11D4C23.81C23C2×C4⋊D4C2×C22.D4C22×C4C2×C4C22
# reps1122331111484

Matrix representation of C24.377C23 ►in GL8(𝔽5)

10000000
01000000
00230000
00430000
00000010
00000001
00001000
00000100
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
12000000
44000000
00100000
00010000
00004000
00000400
00000010
00000001
,
43000000
01000000
00100000
00240000
00004000
00002100
00000040
00000021
,
10000000
01000000
00200000
00020000
00002200
00001300
00000022
00000013

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

C24.377C23 in GAP, Magma, Sage, TeX 

C_2^4._{377}C_2^3
 
% in TeX
 
G:=Group("C2^4.377C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1393);
 
// by ID
 
G=gap.SmallGroup(128,1393);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=c,e*a*e^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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