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G = C23×M4(2)  order 128 = 27

Direct product of C23 and M4(2)

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

Aliases: C23×M4(2), C84C24, C25.9C4, C4.19C25, (C23×C8)⋊16C2, (C2×C8)⋊17C23, (C24×C4).15C2, C2.13(C24×C4), C4.55(C23×C4), (C23×C4).44C4, C4(C22×M4(2)), M4(2)(C22×C4), (C2×C4).694C24, C24.132(C2×C4), (C22×C8)⋊71C22, C22.49(C23×C4), C23.235(C22×C4), (C23×C4).709C22, (C22×C4).1652C23, (C2×C4)2(C2×M4(2)), (C22×C4)(C2×M4(2)), (C2×C4)(C22×M4(2)), (C22×C4).499(C2×C4), (C2×C4).576(C22×C4), (C22×C4)(C22×M4(2)), SmallGroup(128,2302)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C23×M4(2)
C1C2C4C2×C4C22×C4C23×C4C24×C4 — C23×M4(2)
C1C2 — C23×M4(2)
C1C23×C4 — C23×M4(2)
C1C2C2C4 — C23×M4(2)

Generators and relations for C23×M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 988 in 860 conjugacy classes, 732 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4, C4 [×15], C22 [×43], C22 [×56], C8 [×16], C2×C4 [×120], C23 [×43], C23 [×56], C2×C8 [×56], M4(2) [×64], C22×C4 [×140], C24, C24 [×14], C24 [×8], C22×C8 [×28], C2×M4(2) [×112], C23×C4 [×2], C23×C4 [×28], C25, C23×C8 [×2], C22×M4(2) [×28], C24×C4, C23×M4(2)
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], M4(2) [×8], C22×C4 [×140], C24 [×31], C2×M4(2) [×28], C23×C4 [×30], C25, C22×M4(2) [×14], C24×C4, C23×M4(2)

Smallest permutation representation of C23×M4(2)
On 64 points
Generators in S64
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)

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

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

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

80 conjugacy classes

class 1 2A···2O2P···2W4A···4P4Q···4X8A···8AF
order12···22···24···44···48···8
size11···12···21···12···22···2

80 irreducible representations

dim1111112
type++++
imageC1C2C2C2C4C4M4(2)
kernelC23×M4(2)C23×C8C22×M4(2)C24×C4C23×C4C25C23
# reps1228130216

Matrix representation of C23×M4(2) in GL5(𝔽17)

10000
016000
00100
000160
000016
,
160000
01000
00100
00010
00001
,
160000
01000
001600
000160
000016
,
130000
01000
00100
000016
00040
,
160000
016000
001600
00010
000016

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

C23×M4(2) in GAP, Magma, Sage, TeX

C_2^3\times M_4(2)
% in TeX

G:=Group("C2^3xM4(2)");
// GroupNames label

G:=SmallGroup(128,2302);
// by ID

G=gap.SmallGroup(128,2302);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,925,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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