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G = C2×C24.C22  order 128 = 27

Direct product of C2 and C24.C22

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

Aliases: C2×C24.C22, C25.15C22, C24.183C23, C23.171C24, C24.65(C2×C4), (C22×C42)⋊4C2, (C2×C42)⋊82C22, C22.109(C4×D4), (C22×C4).704D4, C23.826(C2×D4), (C23×C4).35C22, C22.62(C23×C4), C23.73(C22×C4), C22.69(C22×D4), C23.360(C4○D4), C22.157(C4⋊D4), (C22×C4).1236C23, C2.C42⋊51C22, C22.73(C4.4D4), C22.31(C42⋊2C2), C22.68(C42⋊C2), C22.100(C22.D4), C2.8(C2×C4×D4), (C22×C4⋊C4)⋊5C2, C2.3(C2×C4⋊D4), (C2×C4⋊C4)⋊99C22, (C2×C22⋊C4)⋊20C4, C22⋊C4⋊35(C2×C4), (C2×C4).672(C2×D4), C2.2(C2×C4.4D4), C2.3(C2×C42⋊2C2), C22.63(C2×C4○D4), (C2×C2.C42)⋊8C2, (C22×C4).380(C2×C4), (C2×C4).205(C22×C4), C2.14(C2×C42⋊C2), C2.5(C2×C22.D4), (C22×C22⋊C4).16C2, (C2×C22⋊C4).414C22, SmallGroup(128,1021)

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

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

Generators and relations for C2×C24.C22
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 844 in 464 conjugacy classes, 196 normal (32 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C25, C2×C2.C42, C24.C22, C22×C42, C22×C22⋊C4, C22×C4⋊C4, C2×C24.C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C23×C4, C22×D4, C2×C4○D4, C24.C22, C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C2×C4.4D4, C2×C42⋊2C2, C2×C24.C22

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

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

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

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

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

56 conjugacy classes

class 1 2A···2O2P2Q2R2S4A···4X4Y···4AJ
order12···222224···44···4
size11···144442···24···4

56 irreducible representations

dim111111122
type+++++++
imageC1C2C2C2C2C2C4D4C4○D4
kernelC2×C24.C22C2×C2.C42C24.C22C22×C42C22×C22⋊C4C22×C4⋊C4C2×C22⋊C4C22×C4C23
# reps12813116816

Matrix representation of C2×C24.C22 ►in GL6(𝔽5)

100000
040000
004000
000400
000040
000004
,
100000
010000
004000
000100
000010
000014
,
100000
010000
004000
000400
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
010000
001000
000100
000010
000001
,
200000
010000
000400
004000
000021
000023
,
400000
040000
000400
001000
000030
000003

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

C2×C24.C22 in GAP, Magma, Sage, TeX 

C_2\times C_2^4.C_2^2
 
% in TeX
 
G:=Group("C2xC2^4.C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1021);
 
// by ID
 
G=gap.SmallGroup(128,1021);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,100]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=e,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

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