Copied to
clipboard

G = C2×C23.25D4order 128 = 27

Direct product of C2 and C23.25D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C23.25D4, C24.140D4, (C22×C8)⋊16C4, C4.3(C22×Q8), C4.47(C23×C4), (C23×C8).18C2, C8.57(C22×C4), C2.D873C22, C4.Q872C22, C4⋊C4.348C23, C23.75(C4⋊C4), (C2×C8).588C23, (C2×C4).185C24, C23.378(C2×D4), (C22×C4).822D4, (C22×C4).103Q8, C4(C23.25D4), C22.82(C4○D8), (C22×C8).565C22, (C23×C4).696C22, C22.132(C22×D4), (C22×C4).1506C23, C42⋊C2.285C22, C4(C2×C2.D8), C4(C2×C4.Q8), (C2×C8)⋊37(C2×C4), C4.85(C2×C4⋊C4), C2.2(C2×C4○D8), (C2×C4.Q8)⋊38C2, (C2×C2.D8)⋊44C2, (C2×C4)2(C2.D8), (C2×C4)2(C4.Q8), C22.35(C2×C4⋊C4), C2.24(C22×C4⋊C4), (C2×C4).239(C2×Q8), (C2×C4).151(C4⋊C4), (C22×C4)(C2.D8), (C22×C4)(C4.Q8), (C2×C4).1567(C2×D4), (C2×C4⋊C4).903C22, (C22×C4).496(C2×C4), (C2×C4).573(C22×C4), (C2×C42⋊C2).53C2, (C2×C4)(C23.25D4), (C2×C4)(C2×C4.Q8), (C2×C4)(C2×C2.D8), (C22×C4)(C2×C2.D8), (C22×C4)(C2×C4.Q8), SmallGroup(128,1641)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C23.25D4
C1C2C22C2×C4C22×C4C23×C4C23×C8 — C2×C23.25D4
C1C2C4 — C2×C23.25D4
C1C22×C4C23×C4 — C2×C23.25D4
C1C2C2C2×C4 — C2×C23.25D4

Generators and relations for C2×C23.25D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 364 in 256 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×26], C2×C4 [×16], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×28], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, C4.Q8 [×8], C2.D8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×8], C42⋊C2 [×4], C22×C8 [×2], C22×C8 [×12], C23×C4, C2×C4.Q8 [×2], C2×C2.D8 [×2], C23.25D4 [×8], C2×C42⋊C2 [×2], C23×C8, C2×C23.25D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C4○D8 [×4], C23×C4, C22×D4, C22×Q8, C23.25D4 [×4], C22×C4⋊C4, C2×C4○D8 [×2], C2×C23.25D4

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

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L4M···4AB8A···8P
order12···222224···444444···48···8
size11···122221···122224···42···2

56 irreducible representations

dim11111112222
type+++++++-+
imageC1C2C2C2C2C2C4D4Q8D4C4○D8
kernelC2×C23.25D4C2×C4.Q8C2×C2.D8C23.25D4C2×C42⋊C2C23×C8C22×C8C22×C4C22×C4C24C22
# reps1228211634116

Matrix representation of C2×C23.25D4 in GL4(𝔽17) generated by

16000
01600
00160
00016
,
16000
01600
0010
001516
,
16000
0100
00160
00016
,
1000
0100
00160
00016
,
16000
01600
0020
0068
,
4000
0100
001616
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,2,6,0,0,0,8],[4,0,0,0,0,1,0,0,0,0,16,0,0,0,16,1] >;

C2×C23.25D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{25}D_4
% in TeX

G:=Group("C2xC2^3.25D4");
// GroupNames label

G:=SmallGroup(128,1641);
// by ID

G=gap.SmallGroup(128,1641);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,352,2804,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

׿
×
𝔽