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G = C24.215C23  order 128 = 27

55th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.215C23, C23.243C24, C22.572- 1+4, C22.772+ 1+4, C2.4D42, C4⋊2(C4×D4), C4⋊C4⋊50D4, C22⋊3(C4×D4), C4⋊D4⋊19C4, C22⋊C4⋊44D4, C2.6(D4⋊6D4), C2.5(Q8⋊5D4), C23.419(C2×D4), C23.18(C22×C4), (C23×C4).56C22, C23.8Q8⋊17C2, C23.323(C4○D4), C23.23D4⋊13C2, C22.134(C23×C4), (C2×C42).437C22, (C22×C4).765C23, C22.114(C22×D4), C24.3C22⋊20C2, C24.C22⋊19C2, (C22×D4).487C22, C23.65C23⋊26C2, C2.C42.523C22, C2.6(C22.47C24), C2.34(C23.33C23), (C2×C4×D4)⋊12C2, (C4×C4⋊C4)⋊43C2, C4⋊C4⋊30(C2×C4), C2.37(C2×C4×D4), C22⋊C4○7(C4⋊C4), (C2×D4)⋊31(C2×C4), (C22×C4⋊C4)⋊11C2, (C4×C22⋊C4)⋊41C2, C22⋊C4⋊16(C2×C4), (C22×C4)⋊35(C2×C4), (C2×C4).889(C2×D4), (C2×C4⋊D4).18C2, (C2×C4).42(C22×C4), (C2×C4).799(C4○D4), (C2×C4⋊C4).827C22, C22.128(C2×C4○D4), (C2×C22⋊C4).442C22, C22⋊C4○3(C2×C4⋊C4), C4⋊C4○3(C2×C22⋊C4), (C2×C22⋊C4)○(C2×C4⋊C4), SmallGroup(128,1093)

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

Derived series
C1 — C22 — C24.215C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C24.215C23
Lower central
C1 — C22 — C24.215C23
Upper central
C1 — C23 — C24.215C23
Jennings
C1 — C23 — C24.215C23

Generators and relations for C24.215C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=ca=ac, g2=a, ab=ba, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 700 in 394 conjugacy classes, 164 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.23D4, C24.C22, C23.65C23, C24.3C22, C22×C4⋊C4, C2×C4×D4, C2×C4×D4, C2×C4⋊D4, C24.215C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×D4, C23.33C23, D42, D4⋊6D4, Q8⋊5D4, C22.47C24, C24.215C23

Smallest permutation representation of C24.215C23
►On 64 points
Generators in S64
(1 33)(2 34)(3 35)(4 36)(5 38)(6 39)(7 40)(8 37)(9 16)(10 13)(11 14)(12 15)(17 56)(18 53)(19 54)(20 55)(21 60)(22 57)(23 58)(24 59)(25 48)(26 45)(27 46)(28 47)(29 52)(30 49)(31 50)(32 51)(41 64)(42 61)(43 62)(44 63)

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

(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)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4T4U···4AH
order12···2222222224···44···4
size11···1222244442···24···4

50 irreducible representations

dim111111111111222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.215C23C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.23D4C24.C22C23.65C23C24.3C22C22×C4⋊C4C2×C4×D4C2×C4⋊D4C4⋊D4C22⋊C4C4⋊C4C2×C4C23C22C22
# reps1112221113116444411

Matrix representation of C24.215C23 ►in GL5(𝔽5)

10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
01000
00100
00010
00001
,
30000
04000
00100
00021
00023
,
30000
01000
00100
00020
00023
,
40000
00100
01000
00010
00001
,
10000
04000
00400
00042
00041

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

C24.215C23 in GAP, Magma, Sage, TeX 

C_2^4._{215}C_2^3
 
% in TeX
 
G:=Group("C2^4.215C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1093);
 
// by ID
 
G=gap.SmallGroup(128,1093);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=c*a=a*c,g^2=a,a*b=b*a,e*d*e^-1=a*d=d*a,g*e*g^-1=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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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