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G = C4⋊M5(2)  order 128 = 27

The semidirect product of C4 and M5(2) acting via M5(2)/C2×C8=C2

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

Aliases: C42M5(2), C42.12C8, C4⋊C1613C2, (C4×C8).38C4, C4.13(C4⋊C8), C8.37(C4⋊C4), C8.41(C2×Q8), (C2×C8).61Q8, (C2×C8).288D4, C8.134(C2×D4), C22.6(C4⋊C8), (C22×C8).31C4, (C2×C42).48C4, C23.37(C2×C8), (C22×C4).15C8, C2.7(C2×M5(2)), (C2×C16).52C22, C42.328(C2×C4), (C2×C8).627C23, (C4×C8).373C22, (C2×C4).82M4(2), C4.65(C2×M4(2)), (C2×M5(2)).23C2, C22.48(C22×C8), (C22×C8).578C22, (C2×C4×C8).66C2, C2.13(C2×C4⋊C8), C4.79(C2×C4⋊C4), (C2×C4).88(C2×C8), (C2×C8).251(C2×C4), (C2×C4).137(C4⋊C4), (C2×C4).612(C22×C4), (C22×C4).487(C2×C4), SmallGroup(128,882)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊M5(2)
C1C2C4C8C2×C8C22×C8C2×C4×C8 — C4⋊M5(2)
C1C22 — C4⋊M5(2)
C1C2×C8 — C4⋊M5(2)
C1C2C2C2C2C4C4C2×C8 — C4⋊M5(2)

Generators and relations for C4⋊M5(2)
 G = < a,b,c | a4=b16=c2=1, bab-1=a-1, ac=ca, cbc=b9 >

Subgroups: 108 in 86 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C16 [×4], C42 [×2], C42 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4 [×2], C4×C8 [×4], C2×C16 [×4], M5(2) [×4], C2×C42, C22×C8 [×2], C4⋊C16 [×4], C2×C4×C8, C2×M5(2) [×2], C4⋊M5(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], M5(2) [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, C2×M5(2) [×2], C4⋊M5(2)

Smallest permutation representation of C4⋊M5(2)
On 64 points
Generators in S64
(1 52 35 30)(2 31 36 53)(3 54 37 32)(4 17 38 55)(5 56 39 18)(6 19 40 57)(7 58 41 20)(8 21 42 59)(9 60 43 22)(10 23 44 61)(11 62 45 24)(12 25 46 63)(13 64 47 26)(14 27 48 49)(15 50 33 28)(16 29 34 51)
(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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8H8I···8T16A···16P
order12222244444···48···88···816···16
size11112211112···21···12···24···4

56 irreducible representations

dim1111111112222
type+++++-
imageC1C2C2C2C4C4C4C8C8D4Q8M4(2)M5(2)
kernelC4⋊M5(2)C4⋊C16C2×C4×C8C2×M5(2)C4×C8C2×C42C22×C8C42C22×C4C2×C8C2×C8C2×C4C4
# reps14124228822416

Matrix representation of C4⋊M5(2) in GL4(𝔽17) generated by

13000
0400
0010
0001
,
0100
8000
0001
0090
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,1,0,0,0,0,0,0,9,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C4⋊M5(2) in GAP, Magma, Sage, TeX

C_4\rtimes M_5(2)
% in TeX

G:=Group("C4:M5(2)");
// GroupNames label

G:=SmallGroup(128,882);
// by ID

G=gap.SmallGroup(128,882);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,1430,102,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^9>;
// generators/relations

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