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G = C2×C22.36C24  order 128 = 27

Direct product of C2 and C22.36C24

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

Aliases: C2×C22.36C24, C22.43C25, C24.485C23, C23.274C24, C42.545C23, C22.772- 1+4, C22.1052+ 1+4, C4⋊Q8⋊76C22, (C4×D4)⋊96C22, (C2×C4).46C24, (C4×Q8)⋊88C22, C4⋊C4.462C23, C22⋊Q8⋊77C22, (C2×D4).292C23, C4.4D4⋊66C22, C22⋊C4.10C23, (C2×Q8).425C23, C42⋊C2⋊90C22, C42⋊2C2⋊24C22, C2.8(C2×2- 1+4), C4⋊D4.217C22, (C23×C4).587C22, (C2×C42).920C22, C2.10(C2×2+ 1+4), (C22×C4).1183C23, (C22×D4).421C22, C22.D4⋊36C22, (C22×Q8).352C22, (C2×C4×D4)⋊75C2, (C2×C4×Q8)⋊47C2, (C2×C4⋊Q8)⋊48C2, C4.74(C2×C4○D4), (C2×C22⋊Q8)⋊65C2, (C2×C4⋊D4).62C2, (C2×C4.4D4)⋊48C2, C2.20(C22×C4○D4), (C2×C42⋊C2)⋊57C2, (C2×C42⋊2C2)⋊33C2, (C2×C4).717(C4○D4), (C2×C4⋊C4).949C22, C22.156(C2×C4○D4), (C2×C22.D4)⋊53C2, (C2×C22⋊C4).532C22, SmallGroup(128,2186)

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

Derived series
C1 — C22 — C2×C22.36C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.36C24
Lower central
C1 — C22 — C2×C22.36C24
Upper central
C1 — C23 — C2×C22.36C24
Jennings
C1 — C22 — C2×C22.36C24

Generators and relations for C2×C22.36C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 844 in 566 conjugacy classes, 396 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C22×Q8, C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C4.4D4, C2×C42⋊2C2, C2×C4⋊Q8, C22.36C24, C2×C22.36C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C25, C22.36C24, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C2×C22.36C24

Smallest permutation representation of C2×C22.36C24
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 11)(6 12)(7 9)(8 10)(13 17)(14 18)(15 19)(16 20)(21 63)(22 64)(23 61)(24 62)(29 37)(30 38)(31 39)(32 40)(33 49)(34 50)(35 51)(36 52)(41 57)(42 58)(43 59)(44 60)(45 53)(46 54)(47 55)(48 56)

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

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

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

(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)

(2 14)(4 16)(6 40)(8 38)(10 30)(12 32)(18 26)(20 28)(21 54)(22 24)(23 56)(33 35)(34 59)(36 57)(41 52)(42 44)(43 50)(45 47)(46 63)(48 61)(49 51)(53 55)(58 60)(62 64)

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

44 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4AD
order12···22···24···44···4
size11···14···42···24···4

44 irreducible representations

dim11111111111244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC2×C22.36C24C2×C42⋊C2C2×C4×D4C2×C4×Q8C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C42⋊2C2C2×C4⋊Q8C22.36C24C2×C4C22C22
# reps111113232116822

Matrix representation of C2×C22.36C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
00040000
00400000
00000100
00001000
00000004
00000040
,
20000000
02000000
00300000
00030000
00000040
00000004
00001000
00000100
,
10000000
04000000
00100000
00040000
00001000
00000100
00000040
00000004
,
10000000
01000000
00400000
00040000
00000400
00001000
00000004
00000010

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;
 

C2×C22.36C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{36}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.36C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2186);
 
// by ID
 
G=gap.SmallGroup(128,2186);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,387,1123,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,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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