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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, C42.542C23, C24.482C23, 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, (C23×C4).584C22, (C2×C42).917C22, (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

Subgroups: 844 in 572 conjugacy classes, 396 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×24], C22, C22 [×10], C22 [×32], C2×C4 [×24], C2×C4 [×48], D4 [×20], Q8 [×4], C23, C23 [×10], C23 [×16], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×56], C22×C4 [×2], C22×C4 [×30], C22×C4 [×12], C2×D4 [×12], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×2], C24, C24 [×2], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×20], C4×D4 [×16], C4⋊D4 [×8], C22⋊Q8 [×24], C22.D4 [×32], C42.C2 [×16], C42⋊2C2 [×16], C23×C4 [×3], C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C22×C4⋊C4, C2×C4×D4 [×2], C2×C4⋊D4, C2×C22⋊Q8, C2×C22⋊Q8 [×2], C2×C22.D4 [×4], C2×C42.C2 [×2], C2×C42⋊2C2 [×2], C22.33C24 [×16], C2×C22.33C24

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ (1+4) [×2], 2- (1+4) [×2], C25, C22.33C24 [×4], C22×C4○D4, C2×2+ (1+4), C2×2- (1+4), C2×C22.33C24

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC2×C22.33C24C22×C4⋊C4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C42.C2C2×C42⋊2C2C22.33C24C23C22C22
# reps1121342216822

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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