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G = M4(2)⋊23D4order 128 = 27

2nd semidirect product of M4(2) and D4 acting through Inn(M4(2))

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

Aliases: M4(2)⋊23D4, C42.682C23, (C8×D4)⋊40C2, C42(C8○D4), C8.87(C2×D4), C4⋊Q8.29C4, C86D435C2, C89D434C2, C4.100(C4×D4), M4(2)(C4⋊C8), C41D4.18C4, C4⋊D4.20C4, C22.20(C4×D4), C4⋊C8.356C22, (C4×M4(2))⋊34C2, (C4×C8).328C22, (C2×C4).654C24, (C2×C8).407C23, C42.211(C2×C4), C4.4D4.16C4, (C4×D4).56C22, C4.200(C22×D4), C23.37(C22×C4), C8⋊C4.156C22, C2.18(Q8○M4(2)), C22⋊C8.231C22, C22.181(C23×C4), (C2×C42).761C22, (C22×C8).435C22, (C22×C4).921C23, (C2×M4(2)).349C22, C22.26C24.26C2, (C2×C4⋊C8)⋊49C2, C2.52(C2×C4×D4), C4⋊C8(C2×M4(2)), (C2×C8○D4)⋊24C2, C2.19(C2×C8○D4), C4⋊C4.161(C2×C4), C4.305(C2×C4○D4), (C2×D4).174(C2×C4), (C2×C4).1086(C2×D4), C22⋊C4.37(C2×C4), (C2×Q8).157(C2×C4), (C22×C8)⋊C231C2, (C2×C4).898(C4○D4), (C2×C4).263(C22×C4), (C22×C4).344(C2×C4), (C2×C4○D4).288C22, SmallGroup(128,1667)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M4(2)⋊23D4
C1C2C4C2×C4C22×C4C2×M4(2)C2×C8○D4 — M4(2)⋊23D4
C1C22 — M4(2)⋊23D4
C1C2×C4 — M4(2)⋊23D4
C1C2C2C2×C4 — M4(2)⋊23D4

Generators and relations for M4(2)⋊23D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a5, ac=ca, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 364 in 246 conjugacy classes, 142 normal (32 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×11], D4 [×18], Q8 [×4], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×12], M4(2) [×4], M4(2) [×10], C22×C4 [×3], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×6], C2×M4(2) [×2], C2×M4(2) [×4], C8○D4 [×8], C2×C4○D4 [×2], C4×M4(2), (C22×C8)⋊C2 [×2], C2×C4⋊C8, C8×D4 [×2], C89D4 [×4], C86D4 [×2], C22.26C24, C2×C8○D4 [×2], M4(2)⋊23D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4, Q8○M4(2), M4(2)⋊23D4

Smallest permutation representation of M4(2)⋊23D4
On 64 points
Generators in S64
(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 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)(17 41)(18 46)(19 43)(20 48)(21 45)(22 42)(23 47)(24 44)(25 58)(26 63)(27 60)(28 57)(29 62)(30 59)(31 64)(32 61)(33 49)(34 54)(35 51)(36 56)(37 53)(38 50)(39 55)(40 52)
(1 25 43 53)(2 26 44 54)(3 27 45 55)(4 28 46 56)(5 29 47 49)(6 30 48 50)(7 31 41 51)(8 32 42 52)(9 61 22 40)(10 62 23 33)(11 63 24 34)(12 64 17 35)(13 57 18 36)(14 58 19 37)(15 59 20 38)(16 60 21 39)
(1 39)(2 36)(3 33)(4 38)(5 35)(6 40)(7 37)(8 34)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)(41 58)(42 63)(43 60)(44 57)(45 62)(46 59)(47 64)(48 61)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4J4K···4P8A···8P8Q···8X
order122222222244444···44···48···88···8
size111122444411112···24···42···24···4

50 irreducible representations

dim11111111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4D4C4○D4C8○D4Q8○M4(2)
kernelM4(2)⋊23D4C4×M4(2)(C22×C8)⋊C2C2×C4⋊C8C8×D4C89D4C86D4C22.26C24C2×C8○D4C4⋊D4C4.4D4C41D4C4⋊Q8M4(2)C2×C4C4C2
# reps11212421284224482

Matrix representation of M4(2)⋊23D4 in GL4(𝔽17) generated by

16000
01600
0090
0088
,
1000
0100
001615
0001
,
01600
1000
0010
0001
,
01600
16000
00139
0044
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,9,8,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,15,1],[0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,16,0,0,0,0,0,13,4,0,0,9,4] >;

M4(2)⋊23D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{23}D_4
% in TeX

G:=Group("M4(2):23D4");
// GroupNames label

G:=SmallGroup(128,1667);
// by ID

G=gap.SmallGroup(128,1667);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,1018,521,124]);
// Polycyclic

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

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