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G = C23.295C24  order 128 = 27

12nd central stem extension by C23 of C24

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

Aliases: C23.295C24, C24.237C23, C23.146(C2×D4), (C22×C4).370D4, C4○(C23.4Q8), C23.16(C4○D4), (C2×C42).28C22, C23.4Q8⋊72C2, C4○2(C23.11D4), C4○3(C23.10D4), (C23×C4).324C22, (C22×C4).498C23, C22.178(C22×D4), C23.11D4⋊140C2, C4○(C23.83C23), C23.10D4.78C2, (C22×D4).495C22, C4○2(C23.81C23), C2.11(C22.19C24), C4.143(C22.D4), C23.81C23⋊146C2, C23.83C23⋊144C2, C2.C42.533C22, C2.10(C22.26C24), C2.13(C23.36C23), (C4×C4⋊C4)⋊52C2, (C2×C4×D4).41C2, (C4×C22⋊C4)⋊48C2, (C2×C4).88(C4○D4), (C2×C4).1560(C2×D4), (C2×C42⋊C2)⋊17C2, (C2×C4⋊C4).838C22, C22.175(C2×C4○D4), (C2×C4)○(C23.4Q8), (C2×C4)○(C23.11D4), C2.12(C2×C22.D4), (C2×C4)○2(C23.10D4), (C2×C22⋊C4).448C22, (C22×C4)○(C23.4Q8), (C2×C4)○(C23.81C23), (C22×C4)○(C23.11D4), (C22×C4)○(C23.10D4), SmallGroup(128,1127)

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

Derived series
C1 — C23 — C23.295C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.295C24
Lower central
C1 — C23 — C23.295C24
Upper central
C1 — C22×C4 — C23.295C24
Jennings
C1 — C23 — C23.295C24

Generators and relations for C23.295C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, eae=ac=ca, faf-1=ad=da, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 500 in 280 conjugacy classes, 108 normal (34 characteristic)
C1, C2, C2, C2, C4, 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, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C22×D4, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.10D4, C23.11D4, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C4×D4, C23.295C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, C2×C22.D4, C22.19C24, C23.36C23, C22.26C24, C23.295C24

Smallest permutation representation of C23.295C24
►On 64 points
Generators in S64
(2 28)(4 26)(6 40)(8 38)(10 30)(12 32)(14 22)(16 24)(17 51)(18 20)(19 49)(33 60)(34 36)(35 58)(41 43)(42 48)(44 46)(45 47)(50 52)(53 55)(54 64)(56 62)(57 59)(61 63)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF
order12···222224···44···4
size11···144441···14···4

44 irreducible representations

dim1111111111222
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D4
kernelC23.295C24C4×C22⋊C4C4×C4⋊C4C23.10D4C23.11D4C23.81C23C23.4Q8C23.83C23C2×C42⋊C2C2×C4×D4C22×C4C2×C4C23
# reps13223111114204

Matrix representation of C23.295C24 ►in GL6(𝔽5)

100000
440000
001000
002400
000010
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
430000
010000
002300
004300
000010
000004
,
200000
330000
003000
000300
000001
000010
,
200000
020000
004000
000400
000030
000003

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

C23.295C24 in GAP, Magma, Sage, TeX 

C_2^3._{295}C_2^4
 
% in TeX
 
G:=Group("C2^3.295C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1127);
 
// by ID
 
G=gap.SmallGroup(128,1127);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=c,g^2=b,a*b=b*a,e*a*e=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=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,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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