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G = C23.514C24  order 128 = 27

231st central stem extension by C23 of C24

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

Aliases: C24.39C23, C23.514C24, C22.2132- 1+4, (Q8×C23)⋊4C2, C23⋊Q8⋊26C2, (C22×C4).397D4, C23.623(C2×D4), C22.55C22≀C2, C23.34D4⋊41C2, (C22×C4).852C23, (C23×C4).417C22, C22.339(C22×D4), (C22×D4).188C22, (C22×Q8).446C22, C23.78C23⋊24C2, C2.C42.242C22, C2.31(C23.38C23), (C2×C4).374(C2×D4), C2.26(C2×C22≀C2), (C2×C4⋊C4).352C22, (C2×C22⋊C4).207C22, (C2×C22.D4).19C2, SmallGroup(128,1346)

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

Derived series
C1 — C23 — C23.514C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C22.D4 — C23.514C24
Lower central
C1 — C23 — C23.514C24
Upper central
C1 — C23 — C23.514C24
Jennings
C1 — C23 — C23.514C24

Generators and relations for C23.514C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 708 in 391 conjugacy classes, 116 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C22×Q8, C22×Q8, C23.34D4, C23⋊Q8, C23.78C23, C2×C22.D4, Q8×C23, C23.514C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, 2- 1+4, C2×C22≀C2, C23.38C23, C23.514C24

Smallest permutation representation of C23.514C24
►On 64 points
Generators in S64
(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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4L4M···4S
order12···2222224···44···4
size11···1222284···48···8

32 irreducible representations

dim11111124
type+++++++-
imageC1C2C2C2C2C2D42- 1+4
kernelC23.514C24C23.34D4C23⋊Q8C23.78C23C2×C22.D4Q8×C23C22×C4C22
# reps134431124

Matrix representation of C23.514C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
00400000
00040000
00000003
00000030
00000300
00003000
,
40000000
04000000
00010000
00100000
00000010
00000001
00004000
00000400
,
10000000
04000000
00100000
00040000
00001000
00000400
00000010
00000004
,
40000000
04000000
00100000
00010000
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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C23.514C24 in GAP, Magma, Sage, TeX 

C_2^3._{514}C_2^4
 
% in TeX
 
G:=Group("C2^3.514C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1346);
 
// by ID
 
G=gap.SmallGroup(128,1346);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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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