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G = C4xM4(2)  order 64 = 26

Direct product of C4 and M4(2)

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

Aliases: C4xM4(2), C4.4C42, C42.9C4, C22.4C42, C42.59C22, C8:6(C2xC4), (C4xC8):13C2, C4o2(C8:C4), C8:C4:12C2, C2.6(C2xC42), C42o(C8:C4), (C2xC8).98C22, C23.27(C2xC4), C4.33(C22xC4), (C22xC4).10C4, (C2xC42).12C2, C2.2(C2xM4(2)), C42o(C2xM4(2)), (C2xC4).142C23, (C2xM4(2)).16C2, C22.18(C22xC4), (C22xC4).105C22, (C2xC4).84(C2xC4), SmallGroup(64,85)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4xM4(2)
C1C2C22C2xC4C22xC4C2xC42 — C4xM4(2)
C1C2 — C4xM4(2)
C1C42 — C4xM4(2)
C1C2C2C2xC4 — C4xM4(2)

Generators and relations for C4xM4(2)
 G = < a,b,c | a4=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 81 in 71 conjugacy classes, 61 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, C23, C42, C42, C2xC8, M4(2), C22xC4, C22xC4, C4xC8, C8:C4, C2xC42, C2xM4(2), C4xM4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, M4(2), C22xC4, C2xC42, C2xM4(2), C4xM4(2)

Smallest permutation representation of C4xM4(2)
On 32 points
Generators in S32
(1 19 26 16)(2 20 27 9)(3 21 28 10)(4 22 29 11)(5 23 30 12)(6 24 31 13)(7 17 32 14)(8 18 25 15)
(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)
(1 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)

G:=sub<Sym(32)| (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)>;

G:=Group( (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19) );

G=PermutationGroup([[(1,19,26,16),(2,20,27,9),(3,21,28,10),(4,22,29,11),(5,23,30,12),(6,24,31,13),(7,17,32,14),(8,18,25,15)], [(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)], [(1,26),(2,31),(3,28),(4,25),(5,30),(6,27),(7,32),(8,29),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19)]])

C4xM4(2) is a maximal subgroup of
M4(2):C8  C42.3Q8  C42.6Q8  C42.26D4  C42.388D4  C42.9Q8  C8:9M4(2)  C82:15C2  C8:6M4(2)  C42.47D4  C42.400D4  C42.401D4  D4:4M4(2)  D4:5M4(2)  Q8:5M4(2)  C42.66D4  C42.405D4  C42.406D4  C42.407D4  C42.408D4  C42.376D4  M4(2):1C8  C8:1M4(2)  C8.6C42  C8:C4:17C4  C42.427D4  C42.430D4  M4(2):12D4  C42.114D4  C42.115D4  M4(2):13D4  M4(2):7Q8  M4(2):8Q8  C42.128D4  C8.5M4(2)  M4(2)o2M4(2)  C42.677C23  C42.259C23  C42.260C23  C42.261C23  M4(2):23D4  M4(2).51D4  M4(2):9Q8  C42.290C23  C42.292C23  C42.294C23  D4:6M4(2)  Q8:6M4(2)  C42.240D4  C42.241D4  C42.242D4  C42.243D4  C42.244D4  M4(2):7D4  M4(2):8D4  M4(2):9D4  M4(2):5Q8  M4(2):6Q8  C42.255D4  C42.256D4  C42.259D4  C42.260D4  C42.261D4  C42.262D4
 D2p.C42: D4.C42  D4.5C42  Dic3:5M4(2)  D10.6C42  Dic7.C42 ...
C4xM4(2) is a maximal quotient of
C8:9M4(2)  C82:15C2  C8:6M4(2)  C23.28C42  C43.C2  C23.17C42  C4:C8:14C4
 C42.D2p: C42.378D4  C43.7C2  Dic3:5M4(2)  D10.6C42  Dic7.C42 ...

40 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P
order1222224···44···48···8
size1111221···12···22···2

40 irreducible representations

dim111111112
type+++++
imageC1C2C2C2C2C4C4C4M4(2)
kernelC4xM4(2)C4xC8C8:C4C2xC42C2xM4(2)C42M4(2)C22xC4C4
# reps1221241648

Matrix representation of C4xM4(2) in GL3(F17) generated by

1300
010
001
,
400
0415
0613
,
100
010
0416
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[4,0,0,0,4,6,0,15,13],[1,0,0,0,1,4,0,0,16] >;

C4xM4(2) in GAP, Magma, Sage, TeX

C_4\times M_4(2)
% in TeX

G:=Group("C4xM4(2)");
// GroupNames label

G:=SmallGroup(64,85);
// by ID

G=gap.SmallGroup(64,85);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,650,117]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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