Copied to
clipboard

G = C42.468C23order 128 = 27

329th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.468C23, C4.482+ 1+4, (C8×D4)⋊21C2, (C4×D8)⋊13C2, D46D48C2, C88D440C2, C4⋊C4.409D4, D4⋊D411C2, D42Q840C2, (C2×D4).238D4, C4.46(C4○D8), (C4×C8).85C22, D4.20(C4○D4), D4.D444C2, C4⋊C4.403C23, C4⋊C8.344C22, (C2×C8).185C23, (C2×C4).495C24, C22⋊C4.107D4, C4.SD1615C2, C23.111(C2×D4), C4⋊Q8.144C22, C2.70(D4○SD16), (C2×D8).139C22, (C4×D4).336C22, (C2×D4).225C23, C23.20D47C2, C4⋊D4.75C22, (C2×Q8).211C23, C2.131(D45D4), C4.Q8.102C22, C2.D8.191C22, C22⋊Q8.75C22, C23.24D428C2, C22⋊C8.204C22, (C22×C8).356C22, Q8⋊C4.13C22, (C2×SD16).97C22, C22.755(C22×D4), D4⋊C4.185C22, C22.49C243C2, (C22×C4).1139C23, C42⋊C2.183C22, C4⋊C4(D4⋊C4), C2.63(C2×C4○D8), C4.220(C2×C4○D4), (C2×C4).924(C2×D4), (C2×C4○D4).201C22, SmallGroup(128,2035)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.468C23
C1C2C4C2×C4C22×C4C2×C4○D4D46D4 — C42.468C23
C1C2C2×C4 — C42.468C23
C1C22C4×D4 — C42.468C23
C1C2C2C2×C4 — C42.468C23

Generators and relations for C42.468C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 400 in 199 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×13], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×2], D4 [×11], Q8 [×5], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8, C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2 [×2], C42⋊C2, C4×D4 [×3], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×C4○D4 [×2], C23.24D4 [×2], C8×D4, C4×D8, D4⋊D4 [×2], D4.D4, C88D4 [×2], D42Q8, C23.20D4 [×2], C4.SD16, D46D4, C22.49C24, C42.468C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C4○D8, D4○SD16, C42.468C23

Smallest permutation representation of C42.468C23
On 64 points
Generators in S64
(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 23 10 33)(2 24 11 34)(3 21 12 35)(4 22 9 36)(5 62 30 28)(6 63 31 25)(7 64 32 26)(8 61 29 27)(13 53 59 39)(14 54 60 40)(15 55 57 37)(16 56 58 38)(17 46 44 51)(18 47 41 52)(19 48 42 49)(20 45 43 50)
(1 56 12 40)(2 55 9 39)(3 54 10 38)(4 53 11 37)(5 42 32 17)(6 41 29 20)(7 44 30 19)(8 43 31 18)(13 34 57 22)(14 33 58 21)(15 36 59 24)(16 35 60 23)(25 52 61 45)(26 51 62 48)(27 50 63 47)(28 49 64 46)
(1 42)(2 43)(3 44)(4 41)(5 60)(6 57)(7 58)(8 59)(9 18)(10 19)(11 20)(12 17)(13 29)(14 30)(15 31)(16 32)(21 46)(22 47)(23 48)(24 45)(25 37)(26 38)(27 39)(28 40)(33 49)(34 50)(35 51)(36 52)(53 61)(54 62)(55 63)(56 64)
(1 38 10 56)(2 37 11 55)(3 40 12 54)(4 39 9 53)(5 46 30 51)(6 45 31 50)(7 48 32 49)(8 47 29 52)(13 36 59 22)(14 35 60 21)(15 34 57 24)(16 33 58 23)(17 62 44 28)(18 61 41 27)(19 64 42 26)(20 63 43 25)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I4J4K4L···4P8A8B8C8D8E···8J
order1222222224···44444···488888···8
size1111444482···24448···822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82+ 1+4D4○SD16
kernelC42.468C23C23.24D4C8×D4C4×D8D4⋊D4D4.D4C88D4D42Q8C23.20D4C4.SD16D46D4C22.49C24C22⋊C4C4⋊C4C2×D4D4C4C4C2
# reps1211212121112114812

Matrix representation of C42.468C23 in GL4(𝔽17) generated by

4000
01300
0010
0001
,
1000
0100
00115
00116
,
0400
4000
00130
00134
,
1000
0100
00116
00146
,
01300
4000
0040
0004
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[0,4,0,0,4,0,0,0,0,0,13,13,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,11,14,0,0,6,6],[0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4] >;

C42.468C23 in GAP, Magma, Sage, TeX

C_4^2._{468}C_2^3
% in TeX

G:=Group("C4^2.468C2^3");
// GroupNames label

G:=SmallGroup(128,2035);
// by ID

G=gap.SmallGroup(128,2035);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,346,248,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽