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G = C24.376C23  order 128 = 27

216th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.376C23, C23.552C24, C22.3272+ 1+4, C22.2432- 1+4, C23.72(C4○D4), (C2×C42).84C22, (C22×C4).162C23, (C23×C4).145C22, C23.Q8.22C2, C23.8Q8.43C2, C23.11D4.30C2, C23.81C23⋊68C2, C23.83C23⋊69C2, C2.50(C22.32C24), C23.63C23⋊119C2, C23.65C23⋊108C2, C2.C42.269C22, C2.33(C22.35C24), C2.59(C22.36C24), C2.49(C22.33C24), C2.104(C23.36C23), (C4×C22⋊C4).73C2, (C2×C4).177(C4○D4), (C2×C4⋊C4).377C22, C22.424(C2×C4○D4), (C2×C22⋊C4).473C22, SmallGroup(128,1384)

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

Derived series
C1 — C23 — C24.376C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.8Q8 — C24.376C23
Lower central
C1 — C23 — C24.376C23
Upper central
C1 — C23 — C24.376C23
Jennings
C1 — C23 — C24.376C23

Generators and relations for C24.376C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=a, g2=b, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×C22⋊C4, C23.8Q8, C23.63C23, C23.65C23, C23.Q8, C23.11D4, C23.81C23, C23.83C23, C24.376C23
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, C24.376C23

Smallest permutation representation of C24.376C23
►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)

(2 12)(4 10)(5 40)(6 8)(7 38)(14 42)(16 44)(17 19)(18 48)(20 46)(22 50)(24 52)(26 54)(28 56)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(45 47)(57 59)(61 63)

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4N4O···4V
order12···22244444···44···4
size11···14422224···48···8

32 irreducible representations

dim1111111112244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC24.376C23C4×C22⋊C4C23.8Q8C23.63C23C23.65C23C23.Q8C23.11D4C23.81C23C23.83C23C2×C4C23C22C22
# reps1123113138422

Matrix representation of C24.376C23 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
20000000
02000000
00030000
00200000
00000010
00000001
00001000
00000100
,
10000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004
,
01000000
10000000
00040000
00400000
00000400
00001000
00000004
00000010
,
40000000
04000000
00200000
00020000
00000100
00001000
00000001
00000010

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

C24.376C23 in GAP, Magma, Sage, TeX 

C_2^4._{376}C_2^3
 
% in TeX
 
G:=Group("C2^4.376C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1384);
 
// by ID
 
G=gap.SmallGroup(128,1384);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,100,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=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,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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