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G = C2×C23.46D4order 128 = 27

Direct product of C2 and C23.46D4

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

Aliases: C2×C23.46D4, C24.181D4, C23.44SD16, C4⋊C4.50C23, C4.Q863C22, C22⋊C867C22, (C2×C8).311C23, (C2×C4).287C24, (C2×D4).77C23, C23.663(C2×D4), (C22×C4).438D4, D4⋊C477C22, C22.35(C2×SD16), C2.13(C22×SD16), C4⋊D4.153C22, (C23×C4).557C22, (C22×C8).348C22, C22.547(C22×D4), C22.121(C8⋊C22), (C22×C4).1004C23, C4.59(C22.D4), (C22×D4).358C22, C22.110(C22.D4), (C2×C4.Q8)⋊34C2, C4.97(C2×C4○D4), (C22×C4⋊C4)⋊34C2, (C2×C22⋊C8)⋊35C2, (C2×C4).847(C2×D4), C2.26(C2×C8⋊C22), (C2×D4⋊C4)⋊39C2, (C2×C4⋊C4)⋊117C22, (C2×C4⋊D4).57C2, (C2×C4).845(C4○D4), C2.52(C2×C22.D4), SmallGroup(128,1821)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C23.46D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C2×C23.46D4
C1C2C2×C4 — C2×C23.46D4
C1C23C23×C4 — C2×C23.46D4
C1C2C2C2×C4 — C2×C23.46D4

Generators and relations for C2×C23.46D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >

Subgroups: 540 in 256 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×30], D4 [×14], C23, C23 [×6], C23 [×12], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×D4, C22×D4, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C4.Q8 [×2], C23.46D4 [×8], C22×C4⋊C4, C2×C4⋊D4, C2×C23.46D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C23.46D4 [×4], C2×C22.D4, C22×SD16, C2×C8⋊C22, C2×C23.46D4

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4N4O4P8A···8H
order12···222222244444···4448···8
size11···122228822224···4884···4

38 irreducible representations

dim111111122224
type++++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4SD16C8⋊C22
kernelC2×C23.46D4C2×C22⋊C8C2×D4⋊C4C2×C4.Q8C23.46D4C22×C4⋊C4C2×C4⋊D4C22×C4C24C2×C4C23C22
# reps112281131882

Matrix representation of C2×C23.46D4 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
180000
0160000
000100
001000
0000160
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
100000
010000
001000
000100
0000160
0000016
,
1320000
140000
004000
0001300
0000125
00001212
,
1600000
1310000
001000
0001600
0000160
000001

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;

C2×C23.46D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{46}D_4
% in TeX

G:=Group("C2xC2^3.46D4");
// GroupNames label

G:=SmallGroup(128,1821);
// by ID

G=gap.SmallGroup(128,1821);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,436,4037,1027,124]);
// Polycyclic

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

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