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G = C23.345C24  order 128 = 27

62nd central stem extension by C23 of C24

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

Aliases: C23.345C24, C24.270C23, C22.1542+ 1+4, C2.18D42, C4⋊C4⋊21D4, (C2×D4).288D4, C23.166(C2×D4), C2.37(D4⋊5D4), C2.13(Q8⋊6D4), C4⋊3(C22.D4), (C23×C4).77C22, C23.23D4⋊40C2, C23.10D4⋊25C2, (C22×C4).512C23, (C2×C42).488C22, C22.225(C22×D4), C24.3C22⋊39C2, (C22×D4).132C22, C23.65C23⋊52C2, C2.C42.102C22, C2.6(C22.53C24), C2.17(C22.26C24), C2.10(C22.34C24), (C2×C4×D4)⋊32C2, (C4×C4⋊C4)⋊55C2, (C2×C4).53(C2×D4), (C2×C4⋊1D4).15C2, (C2×C4).102(C4○D4), (C2×C4⋊C4).227C22, C22.222(C2×C4○D4), (C2×C22.D4)⋊11C2, C2.23(C2×C22.D4), (C2×C22⋊C4).126C22, SmallGroup(128,1177)

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

Derived series
C1 — C23 — C23.345C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.345C24
Lower central
C1 — C23 — C23.345C24
Upper central
C1 — C23 — C23.345C24
Jennings
C1 — C23 — C23.345C24

Generators and relations for C23.345C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=c, g2=b, eae-1=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 740 in 348 conjugacy classes, 112 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22.D4, C4⋊1D4, C23×C4, C22×D4, C22×D4, C4×C4⋊C4, C23.23D4, C23.65C23, C24.3C22, C23.10D4, C2×C4×D4, C2×C22.D4, C2×C4⋊1D4, C23.345C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C22.D4, C22.26C24, C22.34C24, D42, D4⋊5D4, Q8⋊6D4, C22.53C24, C23.345C24

Smallest permutation representation of C23.345C24
►On 64 points
Generators in S64
(1 15)(2 48)(3 13)(4 46)(5 9)(6 42)(7 11)(8 44)(10 18)(12 20)(14 38)(16 40)(17 41)(19 43)(21 29)(22 58)(23 31)(24 60)(25 61)(26 33)(27 63)(28 35)(30 50)(32 52)(34 55)(36 53)(37 45)(39 47)(49 57)(51 59)(54 62)(56 64)

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4V4W4X
order12···22222224···44···444
size11···14444882···24···488

38 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.345C24C4×C4⋊C4C23.23D4C23.65C23C24.3C22C23.10D4C2×C4×D4C2×C22.D4C2×C4⋊1D4C4⋊C4C2×D4C2×C4C22
# reps11213412144122

Matrix representation of C23.345C24 ►in GL6(𝔽5)

010000
100000
004100
000100
000044
000001
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
020000
200000
001400
002400
000030
000003
,
010000
400000
004100
003100
000044
000021
,
400000
040000
001400
002400
000010
000001

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

C23.345C24 in GAP, Magma, Sage, TeX 

C_2^3._{345}C_2^4
 
% in TeX
 
G:=Group("C2^3.345C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1177);
 
// by ID
 
G=gap.SmallGroup(128,1177);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=c,g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,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,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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