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G = C23.315C24  order 128 = 27

32nd central stem extension by C23 of C24

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

Aliases: C23.315C24, C24.250C23, C22.922- 1+4, C4⋊C4⋊34D4, C23.154(C2×D4), (C22×C4).374D4, C2.12(D4⋊6D4), C4.163(C4⋊D4), C2.10(Q8⋊5D4), C23.21(C4○D4), C23.11D4⋊4C2, (C22×C4).50C23, C23.7Q8⋊35C2, (C23×C4).333C22, (C2×C42).464C22, C23.10D4.3C2, C22.195(C22×D4), (C22×Q8).97C22, C24.C22⋊32C2, (C22×D4).503C22, C23.65C23⋊35C2, C23.67C23⋊29C2, C2.22(C22.19C24), C2.C42.79C22, C2.9(C22.50C24), C2.11(C23.38C23), C2.13(C22.46C24), (C2×C4×D4).44C2, (C2×C22⋊Q8)⋊4C2, (C2×C4).680(C2×D4), C2.19(C2×C4⋊D4), (C2×C4).95(C4○D4), (C2×C42⋊C2)⋊21C2, (C2×C4⋊C4).206C22, C22.194(C2×C4○D4), (C2×C22⋊C4).110C22, SmallGroup(128,1147)

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

Derived series
C1 — C23 — C23.315C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.315C24
Lower central
C1 — C23 — C23.315C24
Upper central
C1 — C23 — C23.315C24
Jennings
C1 — C23 — C23.315C24

Generators and relations for C23.315C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=ba=ab, g2=a, ac=ca, ede=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 548 in 302 conjugacy classes, 112 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.7Q8, C24.C22, C23.65C23, C23.65C23, C23.67C23, C23.10D4, C23.11D4, C2×C42⋊C2, C2×C4×D4, C2×C22⋊Q8, C23.315C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4⋊D4, C22.19C24, C23.38C23, D4⋊6D4, Q8⋊5D4, C22.46C24, C22.50C24, C23.315C24

Smallest permutation representation of C23.315C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 37)(6 38)(7 39)(8 40)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 43)(22 44)(23 41)(24 42)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)

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

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

(2 12)(4 10)(5 42)(6 23)(7 44)(8 21)(14 48)(16 46)(17 63)(18 35)(19 61)(20 33)(22 39)(24 37)(26 54)(28 56)(30 58)(32 60)(34 51)(36 49)(38 41)(40 43)(50 62)(52 64)

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111111122224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42- 1+4
kernelC23.315C24C23.7Q8C24.C22C23.65C23C23.67C23C23.10D4C23.11D4C2×C42⋊C2C2×C4×D4C2×C22⋊Q8C4⋊C4C22×C4C2×C4C23C22
# reps112312211244842

Matrix representation of C23.315C24 ►in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001200
000400
000001
000040
,
010000
100000
004000
001100
000040
000004
,
100000
040000
001000
004400
000010
000004
,
100000
010000
003000
002200
000040
000004

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

C23.315C24 in GAP, Magma, Sage, TeX 

C_2^3._{315}C_2^4
 
% in TeX
 
G:=Group("C2^3.315C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1147);
 
// by ID
 
G=gap.SmallGroup(128,1147);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e=g*d*g^-1=a*d=d*a,a*e=e*a,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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