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G = C23.356C24  order 128 = 27

73rd central stem extension by C23 of C24

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

Aliases: C23.356C24, C24.277C23, C22.1632+ 1+4, C2.22D42, C4⋊C4⋊41D4, C22⋊C4⋊38D4, C23⋊2D4⋊14C2, C23.169(C2×D4), C2.42(D4⋊5D4), C2.16(Q8⋊6D4), C23.27(C4○D4), (C23×C4).82C22, C23.Q8⋊13C2, C23.8Q8⋊46C2, C23.23D4⋊41C2, (C22×C4).811C23, (C2×C42).499C22, C22.236(C22×D4), C24.3C22⋊40C2, C24.C22⋊41C2, (C22×D4).515C22, C23.63C23⋊38C2, C2.28(C22.19C24), C2.14(C22.32C24), C2.C42.113C22, C2.20(C22.47C24), (C2×C4×D4)⋊34C2, (C2×C4⋊D4)⋊12C2, (C2×C4).335(C2×D4), (C2×C4).111(C4○D4), (C2×C4⋊C4).238C22, C22.233(C2×C4○D4), (C2×C22.D4)⋊13C2, (C2×C22⋊C4).133C22, SmallGroup(128,1188)

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

Derived series
C1 — C23 — C23.356C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.356C24
Lower central
C1 — C23 — C23.356C24
Upper central
C1 — C23 — C23.356C24
Jennings
C1 — C23 — C23.356C24

Generators and relations for C23.356C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=a, g2=b, ab=ba, 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: 756 in 359 conjugacy classes, 108 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22.D4, C23×C4, C22×D4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C24.3C22, C23⋊2D4, C23.Q8, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C23.356C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C22.32C24, D42, D4⋊5D4, Q8⋊6D4, C22.47C24, C23.356C24

Smallest permutation representation of C23.356C24
►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 17)(2 18)(3 19)(4 20)(5 37)(6 38)(7 39)(8 40)(9 24)(10 21)(11 22)(12 23)(13 52)(14 49)(15 50)(16 51)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)(41 45)(42 46)(43 47)(44 48)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M2N4A···4H4I···4T4U4V4W
order12···22···224···44···4444
size11···14···482···24···4888

38 irreducible representations

dim1111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+4
kernelC23.356C24C23.8Q8C23.23D4C23.63C23C24.C22C24.3C22C23⋊2D4C23.Q8C2×C4×D4C2×C4⋊D4C2×C22.D4C22⋊C4C4⋊C4C2×C4C23C22
# reps1121213121144482

Matrix representation of C23.356C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
340000
320000
001000
000100
000020
000003
,
400000
040000
000100
001000
000002
000030
,
130000
040000
004000
000100
000002
000030
,
300000
030000
004000
000400
000001
000040

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

C23.356C24 in GAP, Magma, Sage, TeX 

C_2^3._{356}C_2^4
 
% in TeX
 
G:=Group("C2^3.356C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1188);
 
// by ID
 
G=gap.SmallGroup(128,1188);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=a,g^2=b,a*b=b*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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