Copied to
clipboard

G = C22×SD32order 128 = 27

Direct product of C22 and SD32

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C22×SD32, C163C23, C8.10C24, Q161C23, D8.1C23, C23.64D8, (C2×C4).95D8, C4.22(C2×D8), C8.55(C2×D4), (C2×C8).263D4, (C22×C16)⋊12C2, (C2×C16)⋊20C22, C2.25(C22×D8), C22.76(C2×D8), C4.16(C22×D4), (C2×C8).572C23, (C22×Q16)⋊14C2, (C2×Q16)⋊49C22, (C22×D8).10C2, (C22×C4).622D4, (C2×D8).150C22, (C22×C8).542C22, (C2×C4).873(C2×D4), SmallGroup(128,2141)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C22×SD32
C1C2C4C8C2×C8C22×C8C22×D8 — C22×SD32
C1C2C4C8 — C22×SD32
C1C23C22×C4C22×C8 — C22×SD32
C1C2C2C2C2C4C4C8 — C22×SD32

Generators and relations for C22×SD32
 G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 500 in 200 conjugacy classes, 100 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C8, C8 [×3], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], C16 [×4], C2×C8 [×6], D8 [×4], D8 [×6], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, C2×C16 [×6], SD32 [×16], C22×C8, C2×D8 [×6], C2×D8 [×3], C2×Q16 [×6], C2×Q16 [×3], C22×D4, C22×Q8, C22×C16, C2×SD32 [×12], C22×D8, C22×Q16, C22×SD32
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, SD32 [×4], C2×D8 [×6], C22×D4, C2×SD32 [×6], C22×D8, C22×SD32

Smallest permutation representation of C22×SD32
On 64 points
Generators in S64
(1 60)(2 61)(3 62)(4 63)(5 64)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 45)(18 46)(19 47)(20 48)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 64)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
(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 17)(2 24)(3 31)(4 22)(5 29)(6 20)(7 27)(8 18)(9 25)(10 32)(11 23)(12 30)(13 21)(14 28)(15 19)(16 26)(33 56)(34 63)(35 54)(36 61)(37 52)(38 59)(39 50)(40 57)(41 64)(42 55)(43 62)(44 53)(45 60)(46 51)(47 58)(48 49)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H8A···8H16A···16P
order12···22222444444448···816···16
size11···18888222288882···22···2

44 irreducible representations

dim1111122222
type+++++++++
imageC1C2C2C2C2D4D4D8D8SD32
kernelC22×SD32C22×C16C2×SD32C22×D8C22×Q16C2×C8C22×C4C2×C4C23C22
# reps111211316216

Matrix representation of C22×SD32 in GL4(𝔽17) generated by

16000
01600
00160
00016
,
16000
0100
00160
00016
,
1000
0100
0071
00167
,
1000
01600
00160
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,7,16,0,0,1,7],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

C22×SD32 in GAP, Magma, Sage, TeX

C_2^2\times {\rm SD}_{32}
% in TeX

G:=Group("C2^2xSD32");
// GroupNames label

G:=SmallGroup(128,2141);
// by ID

G=gap.SmallGroup(128,2141);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

׿
×
𝔽