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G = C2×C4⋊M4(2)  order 128 = 27

Direct product of C2 and C4⋊M4(2)

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

Aliases: C2×C4⋊M4(2), C42.673C23, C4⋊C8⋊79C22, C4⋊3(C2×M4(2)), (C2×C4)⋊11M4(2), (C2×C42).54C4, (C23×C4).40C4, (C22×C4).99Q8, C4.58(C22×Q8), C23.73(C4⋊C4), C4○(C4⋊M4(2)), C42.333(C2×C4), (C2×C8).393C23, (C2×C4).632C24, C24.127(C2×C4), C4.184(C22×D4), (C22×C4).602D4, (C22×C42).31C2, C2.7(C22×M4(2)), C23.222(C22×C4), C22.161(C23×C4), (C23×C4).692C22, (C22×C8).426C22, C22.61(C2×M4(2)), (C22×C4).1500C23, (C2×C42).1103C22, (C22×M4(2)).27C2, (C2×M4(2)).335C22, (C2×C4⋊C8)⋊41C2, C4.60(C2×C4⋊C4), C2.18(C22×C4⋊C4), C22.32(C2×C4⋊C4), (C2×C4).356(C2×Q8), (C2×C4).147(C4⋊C4), (C2×C4).1566(C2×D4), (C2×C4)○(C4⋊M4(2)), (C22×C4).457(C2×C4), (C2×C4).570(C22×C4), SmallGroup(128,1635)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C2×C4⋊M4(2)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C2×C4⋊M4(2)
Lower central
C1 — C22 — C2×C4⋊M4(2)
Upper central
C1 — C22×C4 — C2×C4⋊M4(2)
Jennings
C1 — C2 — C2 — C2×C4 — C2×C4⋊M4(2)

Generators and relations for C2×C4⋊M4(2)
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c5 >

Subgroups: 380 in 288 conjugacy classes, 196 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C4⋊C8, C2×C42, C2×C42, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C2×C4⋊C8, C4⋊M4(2), C22×C42, C22×M4(2), C2×C4⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C2×M4(2), C23×C4, C22×D4, C22×Q8, C4⋊M4(2), C22×C4⋊C4, C22×M4(2), C2×C4⋊M4(2)

Smallest permutation representation of C2×C4⋊M4(2)
►On 64 points
Generators in S64
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB8A···8P
order12···222224···44···48···8
size11···122221···12···24···4

56 irreducible representations

dim1111111222
type++++++-
imageC1C2C2C2C2C4C4D4Q8M4(2)
kernelC2×C4⋊M4(2)C2×C4⋊C8C4⋊M4(2)C22×C42C22×M4(2)C2×C42C23×C4C22×C4C22×C4C2×C4
# reps148121244416

Matrix representation of C2×C4⋊M4(2) ►in GL5(𝔽17)

160000
016000
001600
000160
000016
,
10000
011500
011600
00040
000013
,
10000
016200
00100
00001
00040
,
160000
01000
00100
00010
000016

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,4,0,0,0,0,0,13],[1,0,0,0,0,0,16,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,1,0],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16] >;
 

C2×C4⋊M4(2) in GAP, Magma, Sage, TeX 

C_2\times C_4\rtimes M_4(2)
 
% in TeX
 
G:=Group("C2xC4:M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,1635);
 
// by ID
 
G=gap.SmallGroup(128,1635);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^5>;
 
// generators/relations
 

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