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G = C2×C23.23D4  order 128 = 27

Direct product of C2 and C23.23D4

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

Aliases: C2×C23.23D4, C24.138D4, C25.14C22, C24.532C23, C23.169C24, (C24×C4)⋊1C2, C24⋊8(C2×C4), (C22×C4)⋊51D4, (C22×D4)⋊19C4, C23⋊4(C22×C4), (D4×C23).5C2, C23⋊6(C22⋊C4), (C23×C4)⋊55C22, C23.598(C2×D4), C22.107(C4×D4), C22.60(C23×C4), C22.67(C22×D4), C22.107C22≀C2, C23.358(C4○D4), (C22×C4).447C23, C22.156(C4⋊D4), C2.C42⋊59C22, (C22×D4).462C22, C22.98(C22.D4), C2.6(C2×C4×D4), (C2×C4)⋊20(C2×D4), (C2×D4)⋊36(C2×C4), (C2×C4)⋊5(C22×C4), C2.2(C2×C4⋊D4), (C22×C4)⋊16(C2×C4), C2.3(C2×C22≀C2), C22⋊2(C2×C22⋊C4), (C22×C22⋊C4)⋊4C2, C22.61(C2×C4○D4), C2.8(C22×C22⋊C4), (C2×C22⋊C4)⋊69C22, C2.3(C2×C22.D4), (C2×C2.C42)⋊16C2, SmallGroup(128,1019)

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

Derived series
C1 — C22 — C2×C23.23D4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C2×C23.23D4
Lower central
C1 — C22 — C2×C23.23D4
Upper central
C1 — C24 — C2×C23.23D4
Jennings
C1 — C23 — C2×C23.23D4

Generators and relations for C2×C23.23D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >

Subgroups: 1452 in 788 conjugacy classes, 236 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C23×C4, C22×D4, C22×D4, C25, C2×C2.C42, C23.23D4, C22×C22⋊C4, C22×C22⋊C4, C24×C4, D4×C23, C2×C23.23D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23×C4, C22×D4, C2×C4○D4, C23.23D4, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C4⋊D4, C2×C22.D4, C2×C23.23D4

Smallest permutation representation of C2×C23.23D4
►On 64 points
Generators in S64
(1 60)(2 57)(3 58)(4 59)(5 50)(6 51)(7 52)(8 49)(9 43)(10 44)(11 41)(12 42)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(29 47)(30 48)(31 45)(32 46)(33 64)(34 61)(35 62)(36 63)(37 53)(38 54)(39 55)(40 56)

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

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2W2X2Y2Z2AA4A···4P4Q···4AB
order12···22···222224···44···4
size11···12···244442···24···4

56 irreducible representations

dim1111111222
type++++++++
imageC1C2C2C2C2C2C4D4D4C4○D4
kernelC2×C23.23D4C2×C2.C42C23.23D4C22×C22⋊C4C24×C4D4×C23C22×D4C22×C4C24C23
# reps12831116888

Matrix representation of C2×C23.23D4 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
010000
000400
004000
000042
000001
,
100000
010000
004000
000400
000040
000004
,
100000
040000
004000
000400
000040
000004
,
100000
030000
002000
000200
000042
000001
,
100000
040000
004000
000100
000040
000041

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

C2×C23.23D4 in GAP, Magma, Sage, TeX 

C_2\times C_2^3._{23}D_4
 
% in TeX
 
G:=Group("C2xC2^3.23D4");
 
// GroupNames label
 
G:=SmallGroup(128,1019);
 
// by ID
 
G=gap.SmallGroup(128,1019);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
 
// generators/relations
 

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