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G = C2×C22.33C24  order 128 = 27

Direct product of C2 and C22.33C24

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

Aliases: C2×C22.33C24, C23.19C24, C22.40C25, C24.482C23, C42.542C23, C22.752- 1+4, C22.1032+ 1+4, (C4×D4)⋊94C22, (C2×C4).43C24, C4⋊C4.461C23, C22⋊C4.7C23, C22⋊Q8⋊76C22, (C2×D4).448C23, (C2×Q8).275C23, C42.C2⋊39C22, C42⋊2C2⋊23C22, C2.8(C2×2+ 1+4), C2.6(C2×2- 1+4), C23.260(C4○D4), C4⋊D4.216C22, (C2×C42).917C22, (C23×C4).584C22, (C22×C4).1180C23, (C22×D4).586C22, C22.D4⋊34C22, (C22×Q8).351C22, (C2×C4×D4)⋊73C2, (C22×C4⋊C4)⋊42C2, (C2×C22⋊Q8)⋊64C2, C22.9(C2×C4○D4), (C2×C4⋊C4)⋊130C22, (C2×C4⋊D4).61C2, (C2×C42.C2)⋊38C2, C2.17(C22×C4○D4), (C2×C42⋊2C2)⋊32C2, (C2×C22.D4)⋊51C2, (C2×C22⋊C4).530C22, SmallGroup(128,2183)

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

Derived series
C1 — C22 — C2×C22.33C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C2×C22.33C24
Lower central
C1 — C22 — C2×C22.33C24
Upper central
C1 — C23 — C2×C22.33C24
Jennings
C1 — C22 — C2×C22.33C24

Generators and relations for C2×C22.33C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=c, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 844 in 572 conjugacy classes, 396 normal (20 characteristic)
C1, C2, C2, C2, 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×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C22×C4⋊C4, C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C22⋊Q8, C2×C22.D4, C2×C42.C2, C2×C42⋊2C2, C22.33C24, C2×C22.33C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C25, C22.33C24, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C2×C22.33C24

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

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB
order12···2222222224···44···4
size11···1222244442···24···4

44 irreducible representations

dim111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC2×C22.33C24C22×C4⋊C4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C42.C2C2×C42⋊2C2C22.33C24C23C22C22
# reps1121342216822

Matrix representation of C2×C22.33C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
02000000
30000000
00020000
00300000
00004000
00000100
00000040
00000001
,
30000000
03000000
00300000
00030000
00000004
00000040
00000400
00004000
,
01000000
10000000
00040000
00400000
00000010
00000001
00004000
00000400
,
40000000
04000000
00100000
00010000
00000100
00001000
00000001
00000010

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

C2×C22.33C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{33}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.33C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2183);
 
// by ID
 
G=gap.SmallGroup(128,2183);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,184,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=c,f^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,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=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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